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Journal articles on the topic 'Tychonoff space'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

ALZahrani, Samirah. "C-Tychonoff and L-Tychonoff Topological Spaces." European Journal of Pure and Applied Mathematics 11, no. 3 (July 31, 2018): 882–92. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3253.

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A topological space X is called C-Tychonoff if there exist a one-to-one function f from X onto a Tychonoff space Y such that f restriction K from K onto f(K) is a homeomorphism for each compact subspace K of X. We discuss this property and illustrate the relationships between C-Tychonoffness and some other properties like submetrizability, local compactness, L-Tychononess, C-normality, C-regularity, epinormality, sigma-compactness, pseudocompactness and zero-dimensional.

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2

Shakhmatov, Dimitri, Mikhail Tkachenko, Vladimir V. Tkachuk, and Richard G. Wilson. "Strengthening connected Tychonoff topologies." Applied General Topology 3, no. 2 (October 1, 2002): 113. http://dx.doi.org/10.4995/agt.2002.2058.

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<p>The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N<sub>0</sub> admits a finer strongly σ-discrete connected Tychonoff topology of weight 2<sup>c</sup>. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological group.</p>

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3

Tzannes, V. "A Tychonoff non-normal space." International Journal of Mathematics and Mathematical Sciences 16, no. 3 (1993): 615–16. http://dx.doi.org/10.1155/s0161171293000754.

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A Tychonoff non-normal space is constructed which can be used for the construction of a regular space on which every weakly continuous (hence everyθ-continuous orη-continuous) map into a given space is constant.

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4

Rice, Michael D. "Reflexive objects in topological categories." Mathematical Structures in Computer Science 6, no. 4 (August 1996): 375–86. http://dx.doi.org/10.1017/s0960129500001079.

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This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.

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5

Basu, C. K., and S. S. Mandal. "Maximal Tychonoff spaces and normal isolator covers." Publications de l'Institut Math?matique (Belgrade) 99, no. 113 (2016): 217–25. http://dx.doi.org/10.2298/pim1613217b.

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We introduce a new kind of cover called a normal isolator cover to characterize maximal Tychonoff spaces. Such a study is used to provide an alternative proof of an interesting result of Feng and Garcia-Ferreira in 1999 that every maximal Tychonoff space is extremally disconnected. Maximal tychonoffness of subspaces is also discussed.

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6

Gabriyelyan, S. S. "Free Locally Convex Spaces and the k-space Property." Canadian Mathematical Bulletin 57, no. 4 (December 1, 2014): 803–9. http://dx.doi.org/10.4153/cmb-2014-019-7.

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AbstractLet L(X) be the free locally convex space over a Tychonoff space X. Then L(X) is a k-space if and only if X is a countable discrete space. We prove also that L(D) has uncountable tightness for every uncountable discrete space D.

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7

Çoker, Doğan, A. Haydar Eş, and Necla Turanli. "A Tychonoff theorem in intuitionistic fuzzy topological spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 70 (2004): 3829–37. http://dx.doi.org/10.1155/s0161171204403603.

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The purpose of this paper is to prove a Tychonoff theorem in the so-called “intuitionistic fuzzy topological spaces.” After giving the fundamental definitions, such as the definitions of intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic fuzzy topological space, fuzzy continuity, fuzzy compactness, and fuzzy dicompactness, we obtain several preservation properties and some characterizations concerning fuzzy compactness. Lastly we give a Tychonoff-like theorem.

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8

Jumaev, D. "(O-C)-compact Spaces and Hyperspaces Functor." Bulletin of Science and Practice 5, no. 4 (April 15, 2019): 30–37. http://dx.doi.org/10.33619/2414-2948/41/03.

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In the work, it is established that the space of all nonempty compact subsets of a Tychonoff space is (O-C)–compact if and only if the give Tychonoff space is (O-C)–compact. Further, for a map f:X→Y the map expβX→Y is (O-C)–compact if and only if the map f is (O-C)–compact.

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9

Kocinac, Ljubisa. "On spaces of group-valued functions." Filomat 25, no. 2 (2011): 163–72. http://dx.doi.org/10.2298/fil1102163k.

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10

Elfard, Ali Sayed. "Neighborhood base at the identity of free paratopological groups." Topological Algebra and its Applications 1 (November 12, 2013): 31–36. http://dx.doi.org/10.2478/taa-2013-0004.

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AbstractIn 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.

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11

DIMOV, GEORGI D., and GINO TIRONI. "Hausdorff compactifications and zero-one measures." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 3 (November 2001): 495–505. http://dx.doi.org/10.1017/s0305004101005461.

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It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1, 2, 4, 12]). Using the technique developed in [5, 6], we find for any Tychonoff space X a Boolean algebra [Bscr ]X and a set [Lscr ]X of sublattices of [Bscr ]X having the following property: for any Hausdorff compactification cX of X there exists a (unique) LcX ∈ [Lscr ]X such that the maximal spectrum of LcX and the space of all u-regular zero-one measures on the Boolean subalgebra b(LcX) of [Bscr ]X, generated by LcX, are Hausdorff compactifications of X equivalent to cX. Let us give more details now.

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12

Pestov, Vladimir. "A remark on embedding topological groups into products." Bulletin of the Australian Mathematical Society 49, no. 3 (June 1994): 519–21. http://dx.doi.org/10.1017/s0004972700016622.

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Let P be a class of topological groups such that every topological group is isomorphic to a topological subgroup of the direct product (with Tychonoff topology) of a subfamily of P. Then every Tychonoff space is homeomorphic to a subspace of a group from P.

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13

DUBE, THEMBA. "A NOTE ON RELATIVE PSEUDOCOMPACTNESS IN THE CATEGORY OF FRAMES." Bulletin of the Australian Mathematical Society 87, no. 1 (May 22, 2012): 120–30. http://dx.doi.org/10.1017/s000497271200024x.

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AbstractA subspace S of Tychonoff space X is relatively pseudocompact in X if every f∈C(X) is bounded on S. As is well known, this property is characterisable in terms of the functor υ which reflects Tychonoff spaces onto the realcompact ones. A device which exists in the category CRegFrm of completely regular frames which has no counterpart in Tych is the functor which coreflects completely regular frames onto the Lindelöf ones. In this paper we use this functor to characterise relative pseudocompactness.

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14

Cao, Jiling. "On isocompactness of function spaces." Bulletin of the Australian Mathematical Society 60, no. 3 (December 1999): 483–86. http://dx.doi.org/10.1017/s0004972700036649.

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Let Cp(X) be the space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology. In this note, we show that if X is a space, then Cp(X) is isocompact. This gives an answer to a recent question of Arkhangel'skii in the class of spaces.

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15

Smrekar, Jaka. "Homotopy Characterization of ANR Function Spaces." Journal of Function Spaces and Applications 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/925742.

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LetYbe an absolute neighbourhood retract (ANR) for the class of metric spaces and letXbe a topological space. LetYXdenote the space of continuous maps fromXtoYequipped with the compact open topology. We show that ifXis a compactly generated Tychonoff space andYis not discrete, thenYXis an ANR for metric spaces if and only ifXis hemicompact andYXhas the homotopy type of a CW complex.

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16

Prasetyo, Puguh Wahyu, Dian Ariesta Yuwaningsih, and Burhanudin Arif Nurnugroho. "Algebraic Structure of Supernilpotent Radical Class Constructed from a Topology Thychonoff Space." Al-Jabar : Jurnal Pendidikan Matematika 11, no. 2 (December 19, 2020): 331–38. http://dx.doi.org/10.24042/ajpm.v11i2.6897.

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A radical class of rings is called a supernilpotent radicals if it is hereditary and it contains the class for some positive integer In this paper, we start by exploring the concept of Tychonoff space to build a supernilpotent radical. Let be a Tychonoff space that does not contain any isolated point. The set of all continuous real-valued functions defined on is a prime essential ring. Finally, we can show that the class of rings is a supernilpotent radical class containing the matrix ring .

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17

Just, Winfried, and Jamal Tartir. "A $\kappa $-normal, not densely normal Tychonoff space." Proceedings of the American Mathematical Society 127, no. 3 (1999): 901–5. http://dx.doi.org/10.1090/s0002-9939-99-04587-6.

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18

Srivastava, Kavita. "On the Stone-Čeck compactification of an orbit space." Bulletin of the Australian Mathematical Society 36, no. 3 (December 1987): 435–39. http://dx.doi.org/10.1017/s0004972700003725.

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By extending the given action of a discrete group G on a Tychonoff space X to βX, it is proved that the Stone-Čech compactification of the orbit space of X is the orbit space of the Stone-Čech compactification βX of X, when G is finite. The notion of G-retractive spaces is introduced and it is proved that the orbit space of a G-retractive space with G finite, is G-retractive.

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19

SONG, YANKUI. "A PSEUDOCOMPACT TYCHONOFF SPACE THAT IS NOT STAR LINDELÖF." Bulletin of the Australian Mathematical Society 84, no. 3 (July 21, 2011): 452–54. http://dx.doi.org/10.1017/s0004972711002413.

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AbstractLet P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X, there exists a subspace A⊆X with property P such that X=St(A,𝒰), where St(A,𝒰)=⋃ {U∈𝒰:U∩A≠0̸}. In this paper we construct an example of a pseudocompact Tychonoff space that is not star Lindelöf, which gives a negative answer to Alas et al. [‘Countability and star covering properties’, Topology Appl.158 (2011), 620–626, Question 3].

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20

Azad, K. K., and Gunjan Agrawal. "On the Projective Cover of an orbit space." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 2 (April 1989): 308–12. http://dx.doi.org/10.1017/s1446788700030780.

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AbstractIn this paper, we obtain the projective cover of the orbit space X/G in terms of the orbit space of the projective space of X, when X is a Tychonoff G-space and G is a finite discrete group. An example shows that finiteness of G is needed.

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21

Ramkumar, S., and C. Ganesa Moorthy. "EXTENDABILITY OF SEMI-METRICS TO COMPACTIFICATIONS." Asian-European Journal of Mathematics 06, no. 01 (March 2013): 1350015. http://dx.doi.org/10.1142/s1793557113500150.

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The extendability of a continuous semi-metric on a Tychonoff space to a compactification of the space is discussed. The extended semi-metrics are continuous on compactifications, constructed through an axiomatic approach.

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22

Karapınar, Dünya. "A compactification of an orbit space." Filomat 32, no. 10 (2018): 3429–34. http://dx.doi.org/10.2298/fil1810429k.

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Let X be a Tychonoff G-space, G be a finite discrete group and A be a dense and invariant subspace of X. In this paper, by means of Gelfand?s method, we construct a compactification of the orbit space A/G. As an application, we show that the set of maximal ideals of even function ring with Stone topology is a compactification of non-negative rationals.

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23

Kocinac, Ljubisa D. R. "Closure properties of function spaces." Applied General Topology 4, no. 2 (October 1, 2003): 255. http://dx.doi.org/10.4995/agt.2003.2030.

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24

SPADARO, SANTI. "P-SPACES AND THE VOLTERRA PROPERTY." Bulletin of the Australian Mathematical Society 87, no. 2 (July 31, 2012): 339–45. http://dx.doi.org/10.1017/s0004972712000585.

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AbstractWe study the relationship between generalisations ofP-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two denseGδhave dense (nonempty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almostP-space is Volterra and that there are Tychonoff nonweakly Volterra weakP-spaces. These results should be compared with the fact that everyP-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volterra. We also show an example of a Hausdorff space which contains a nonweakly Volterra subspace and is both a weakP-space and an almostP-space.

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25

SONG, YAN-KUI. "REMARKS ON QUASI-LINDELÖF SPACES." Bulletin of the Australian Mathematical Society 88, no. 3 (June 12, 2013): 506–8. http://dx.doi.org/10.1017/s0004972713000385.

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AbstractIn this paper, we show that there exist a Tychonoff quasi-Lindelöf space $X$ and a compact space $Y$ such that $X\times Y$ is not quasi-Lindelöf. This answers negatively an open question of Petra Staynova.

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26

Bainov, D. D., S. I. Kostadinov, and P. P. Zabreiko. "Lp-equivalence of a linear and a nonlinear impulsive differential equation in a Banach space." Proceedings of the Edinburgh Mathematical Society 36, no. 1 (February 1993): 17–33. http://dx.doi.org/10.1017/s0013091500005861.

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27

Gil, Mariusz, and Stanisław Wędrychowicz. "Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity." Journal of Function Spaces and Applications 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/692879.

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We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour. The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space, Fréchet space, and Sobolew space.

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28

BEZHANISHVILI, GURAM, NICK BEZHANISHVILI, JOEL LUCERO-BRYAN, and JAN VAN MILL. "TYCHONOFF HED-SPACES AND ZEMANIAN EXTENSIONS OF S4.3." Review of Symbolic Logic 11, no. 1 (January 14, 2018): 115–32. http://dx.doi.org/10.1017/s1755020317000314.

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29

Ferrando, Juan Carlos. "A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X." Journal of Function Spaces 2018 (September 10, 2018): 1–5. http://dx.doi.org/10.1155/2018/8219246.

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We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

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30

AlZahrani, Samirah, and Nadia Gheith. "On Epicompletely Regularity." Nanoscience and Nanotechnology Letters 12, no. 2 (February 1, 2020): 263–69. http://dx.doi.org/10.1166/nnl.2020.3086.

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A topological space (X, τ) is said to be epicompletely regular if there is a coarser topological space (X, τ′) that is Tychonoff. In this study we investigated this natural type of complete regularity. We give some examples to explain some relationships between epicompletely regular space and some separation axioms.

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31

Kundu, S., and R. A. McCoy. "Topologies between compact and uniform convergence on function spaces." International Journal of Mathematics and Mathematical Sciences 16, no. 1 (1993): 101–9. http://dx.doi.org/10.1155/s0161171293000122.

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32

Pestov, Vladimir. "Free Abelian topological groups and the Pontryagin-Van Kampen duality." Bulletin of the Australian Mathematical Society 52, no. 2 (October 1995): 297–311. http://dx.doi.org/10.1017/s0004972700014726.

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We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

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33

Aull, C. E. "Some embeddings related to C*-embeddings." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 1 (February 1988): 88–104. http://dx.doi.org/10.1017/s1446788700031396.

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AbstractA space S is R*-embedded (G*-embedded) in a space X if two disjoint regular closed sets (closure disjoint open sets) of S are contained in disjoint regular closed sets (extended to closure disjoint open sets) of X. A space S is R-extendable to a space X if any regular closed set of S can be extended to a regular closed set of X. It is shown that R*-embedding and G*-embedding are identical with C*-embedding for certain fairly general classes of Tychonoff spaces. Under certain conditions it is shown that R-extendability is related to z-embedding. Spaces in which the regular open sets are C and C*-embedded are also investigated.

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34

Baron, Karol. "On Orthogonally Additive Functions With Big Graphs." Annales Mathematicae Silesianae 31, no. 1 (September 26, 2017): 57–62. http://dx.doi.org/10.1515/amsil-2016-0016.

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Abstract Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

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35

Basu, C. K., M. K. Ghosh, and S. S. Mandal. "A generalization of $ H $-closed spaces." Tamkang Journal of Mathematics 39, no. 2 (June 30, 2008): 143–54. http://dx.doi.org/10.5556/j.tkjm.39.2008.25.

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Whereas a space $ X $ can be embedded in a compact space if and only if it is Tychonoff, every space $ X $ can be embedded in an $H$-closed space(a generalization of compact space). In this paper, we further generalize, the concept of $H$-closedness into $ gH $-closedness and have shown that every connected space is either a $ gH $-closed space or can be embedded in a $ gH $-closed space. Also, in a locally connected regular space the concept of $ gH $-closedness is equivalent to the concepts of $ J $-ness and strong $ J $-ness due to E. Michael [7] and $ \theta $J-ness due to C.K. Basu et. al [1]. Several characterizations and properties of $ gH $-closed spaces with respect to subspaces, products and functional preservations (along with various examples) are given.

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36

Dahmen, Rafael, and Gábor Lukács. "Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts." Axioms 10, no. 3 (July 19, 2021): 155. http://dx.doi.org/10.3390/axioms10030155.

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The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

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Bella, Angelo, and Rodrigo Hernández-Gutiérrez. "A non-discrete space X with Cp(X) Menger at infinity." Applied General Topology 20, no. 1 (April 1, 2019): 223. http://dx.doi.org/10.4995/agt.2019.10714.

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<p>In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of C<sub>p</sub>(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.</p>

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38

Ferrando, J. C., and M. López-Pellicer. "Covering properties of Cp(X) and Ck(X)." Filomat 34, no. 11 (2020): 3575–99. http://dx.doi.org/10.2298/fil2011575f.

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Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when Cp (X) or Ck (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X).

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39

Negri, Sara, and Silvio Valentini. "Tychonoff's theorem in the framework of formal topologies." Journal of Symbolic Logic 62, no. 4 (December 1997): 1315–32. http://dx.doi.org/10.2307/2275645.

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In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

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40

Lyakhovets, Daniil, and Alexander Osipov. "Selection principles and games in bitopological function spaces." Filomat 33, no. 14 (2019): 4535–40. http://dx.doi.org/10.2298/fil1914535l.

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For a Tychonoff space X, we denote by (C(X), ?k ?p) the bitopological space of all real-valued continuous functions on X, where ?k is the compact-open topology and ?p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X),?k,?p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X),?k,?p).

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41

Çetkin, Vildan. "On measures of parameterized fuzzy compactness." Filomat 34, no. 9 (2020): 2927–38. http://dx.doi.org/10.2298/fil2009927c.

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In the present study, the parameterized degree of compactness of a lattice valued fuzzy soft set is described in a fuzzy soft topological space. The extended versions of the basic compactness properties known in general topology are investigated for the given notion and some further characterizations of parameterized degree of fuzzy compactness are specified. In addition, a generalized version of Tychonoff Theorem is proved in the product fuzzy soft topological space.

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42

Kim, Junhui. "A non-2-starcompact, pseudocompact Tychonoff space whose hyperspace is 2-starcompact." Topology and its Applications 160, no. 1 (January 2013): 126–32. http://dx.doi.org/10.1016/j.topol.2012.10.009.

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43

GABRIYELYAN, SAAK S., and SIDNEY A. MORRIS. "SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL." Bulletin of the Australian Mathematical Society 97, no. 1 (August 7, 2017): 110–18. http://dx.doi.org/10.1017/s0004972717000673.

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For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

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Jayanthan, A. J., and V. Kannan. "Joins of topologies homeomorphic to the rationals." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 47, no. 2 (October 1989): 256–62. http://dx.doi.org/10.1017/s1446788700031682.

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AbstractLet Q be the space of all rational numbers and (X, τ) be a topological space where X is countably infinite. Here we prove that (1) τ is the join of two topologies on X both homeomorphic to Q if and only if τ is non-compact and metrizable, and (2) τ is the join of topologies on X each homeomorphic to Q if and only if τ is Tychonoff and noncompact.

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Arhangel’skii, A. V., and S. Tokgöz. "The unions of dense metrizable subspaces with certain local properties." Filomat 29, no. 1 (2015): 83–88. http://dx.doi.org/10.2298/fil1501083a.

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Many important examples of topological spaces can be represented as a union of a finite or countable collection of metrizable subspaces. However, it is far from clear which spaces in general can be obtained in this way. Especially interesting is the case when the subspaces are dense in the union. We present below several results in this direction. In particular, we show that if a Tychonoff space X is the union of a countable family of dense metrizable locally compact subspaces, then X itself is metrizable and locally compact. We also prove a similar result for metrizable locally separable spaces. Notice in this connection that the union of two dense metrizable subspaces needn?t be metrizable. Indeed, this is witnessed by a well-known space constructed by R.W. Heath.

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Parsinia, Mehdi. "R-P-spaces and subrings of C(X)." Filomat 32, no. 1 (2018): 319–28. http://dx.doi.org/10.2298/fil1801319p.

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A Tychonoff space X is called a P-space if Mp = Op for each p ? ?X. For a subring R of C(X), we call X an R-P-space, if Mp ? R = Op ? R for each p ? ?X. Various characterizations of R-P-spaces are investigated some of which follows from constructing the smallest invertible subring of C(X) in which R is embedded, S-1R R. Moreover, we study R-P-spaces when R is an intermediate ring or an intermediate C-ring. We follow a new approach to some results of [W. Murray, J. Sack and S. Watson, P-spaces and intermediate rings of continuous functions, Rocky Mount. J. Math., to appear]. Also, some algebraic characterizations of P-spaces via intermediate rings are given. Finally, we establish some characterizations of C(X) among intermediate C-rings which are of the form I + C*(X), where I is an ideal in C(X).

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Nsayi, Jissy Nsonde. "Natural extension of EF-spaces and EZ-spaces to the pointfree context." Mathematica Slovaca 69, no. 5 (October 25, 2019): 979–88. http://dx.doi.org/10.1515/ms-2017-0282.

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Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

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Abd-Allah, Ahmed Saeed. "Characterized Fuzzy R2.5 and Characterized Fuzzy T3.5 Spaces." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 1 (March 30, 2017): 7048–73. http://dx.doi.org/10.24297/jam.v13i1.5684.

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This paper, deals with, introduce and study the notions of haracterized fuzzy R2.5 spaces and of characterized fuzzy T3.5 spaces by using the notion of fuzzy function family presented in [21] and the notion of Ï†1,2Ïˆ1,2-fuzzy continuous mappings presented in [5] as a generalization of all the weaker and stronger forms of the fuzzy completely regular spaces introduced in [11,24,26,29]. We denote by characterized fuzzy T3.5 space or characterized fuzzy Tychonoff space to the characterized fuzzy space which is characterized fuzzy T1 and characterized fuzzy R2.5 space in this sense. The relations between the characterized fuzzy T3.5 spaces, the characterized fuzzy T4 spaces and the characterized fuzzy T3 spaces are introduced. When the given fuzzy topological space is normal, then the related characterized fuzzy space is finer than the associated characterized fuzzy proximity space which is presented in [1]. Moreover, the associated characterized fuzzy proximity spaces and the characterized fuzzy T4 spaces are identical with help of the complementarilysymmetric fuzzy topogenous structure, that is, identified with the fuzzy proximity Î´. More generally, the fuzzy function family of all Ï†1,2Ïˆ1,2-fuzzy continuous mappings are applied to show that the characterized fuzzy R2.5 spaces and the associated characterized fuzzy proximity spaces are identical.

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Gabriyelyan, Saak S., and Sidney A. Morris. "Free Subspaces of Free Locally Convex Spaces." Journal of Function Spaces 2018 (2018): 1–5. http://dx.doi.org/10.1155/2018/2924863.

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IfXandYare Tychonoff spaces, letL(X)andL(Y)be the free locally convex space overXandY, respectively. For generalXandY, the question of whetherL(X)can be embedded as a topological vector subspace ofL(Y)is difficult. The best results in the literature are that ifL(X)can be embedded as a topological vector subspace ofL(I), whereI=[0,1], thenXis a countable-dimensional compact metrizable space. Further, ifXis a finite-dimensional compact metrizable space, thenL(X)can be embedded as a topological vector subspace ofL(I). In this paper, it is proved thatL(X)can be embedded inL(R)as a topological vector subspace ifXis a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case ifX=Rn, n∈N.It is also shown that ifGandQdenote the Cantor space and the Hilbert cubeIN, respectively, then (i)L(X)is embedded inL(G)if and only ifXis a zero-dimensional metrizable compact space; (ii)L(X)is embedded inL(Q)if and only ifYis a metrizable compact space.

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Hager, Anthony W., and Jorge Martinez. "Fraction-Dense Algebras and Spaces." Canadian Journal of Mathematics 45, no. 5 (October 1, 1993): 977–96. http://dx.doi.org/10.4153/cjm-1993-054-6.

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AbstractA fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense ƒ- rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover. R-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density.

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