On the maximalG-compactification of products of twoG-spaces

AbstractAn internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space $$C_{p}(X)$$ C p ( X ) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in $$\mathbb {R}^{X}$$ R X if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is $$\sigma $$ σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for $$\sigma $$ σ -boundedness of X is shown.

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

10.1017/s0004972717000673 ◽

2017 ◽

Vol 97 (1) ◽

pp. 110-118 ◽

Author(s):

SAAK S. GABRIYELYAN ◽

SIDNEY A. MORRIS

Keyword(s):

Topological Group ◽

Vector Space ◽

Topological Vector Space ◽

Unit Interval ◽

Vector Subspace ◽

Locally Compact ◽

Tychonoff Space ◽

Abelian Topological Group ◽

Free Abelian Topological Group ◽

Usual Topology

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.

The Maximal Extension of a Zero-dimensional Product Space

Canadian Mathematical Bulletin ◽

10.4153/cmb-1983-031-9 ◽

1983 ◽

Vol 26 (2) ◽

pp. 192-201

Author(s):

Haruto Ohta

Keyword(s):

Product Space ◽

Topological Property ◽

Continuous Extension ◽

Dense Subspace ◽

Hausdorff Spaces ◽

Maximal Extension ◽

Tychonoff Space ◽

Continuous Map ◽

Countable Compactness ◽

AbstractIt is known that if a topological property of Tychonoff spaces is closed-hereditary, productive and possessed by all compact Hausdorff spaces, then each (0-dimensional) Tychonoff space X is a dense subspace of a (0-dimensional) Tychonoff space with such that each continuous map from X to a (0-dimensional) Tychonoff space with admits a continuous extension over . In response to Broverman's question [Canad. Math. Bull. 19 (1), (1976), 13–19], we prove that if for every two 0-dimensional Tychonoff spaces X and Y, if and only if , then is contained in countable compactness.

A remark on embedding topological groups into products

10.1017/s0004972700016622 ◽

1994 ◽

Vol 49 (3) ◽

pp. 519-521

Author(s):

Vladimir Pestov

Keyword(s):

Topological Group ◽

Direct Product ◽

Topological Groups ◽

Tychonoff Space ◽

Tychonoff Topology

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.

A NOTE ON RELATIVE PSEUDOCOMPACTNESS IN THE CATEGORY OF FRAMES

10.1017/s000497271200024x ◽

2012 ◽

Vol 87 (1) ◽

pp. 120-130 ◽

Author(s):

THEMBA DUBE

Keyword(s):

Tychonoff Space ◽

Completely Regular ◽

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.

Strengthening connected Tychonoff topologies

Applied General Topology ◽

10.4995/agt.2002.2058 ◽

2002 ◽

Vol 3 (2) ◽

pp. 113 ◽

Author(s):

Dimitri Shakhmatov ◽

Mikhail Tkachenko ◽

Vladimir V. Tkachuk ◽

Richard G. Wilson

Keyword(s):

Topological Group ◽

Continuous Image ◽

Tychonoff Space ◽

Tychonoff Topology

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) ≤ N0 admits a finer strongly σ-discrete connected Tychonoff topology of weight 2c. 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.

Local compactness in free topological groups

10.1017/s0004972700037631 ◽

2003 ◽

Vol 68 (2) ◽

pp. 243-265 ◽

Cited By ~ 4

Author(s):

Peter Nickolas ◽

Mikhail Tkachenko

Keyword(s):

Topological Group ◽

Compact Set ◽

Topological Groups ◽

Locally Compact ◽

Tychonoff Space ◽

Abelian Topological Group ◽

Local Compactness ◽

Countable Type ◽

Free Abelian Topological Group ◽

Countable Character

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

Maximal Tychonoff spaces and normal isolator covers

Publications de l Institut Mathematique ◽

10.2298/pim1613217b ◽

2016 ◽

Vol 99 (113) ◽

pp. 217-225

Author(s):

C.K. Basu ◽

S.S. Mandal

Keyword(s):

Alternative Proof ◽

Tychonoff Space ◽

Extremally Disconnected ◽

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.

General rings of functions

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700020711 ◽

1975 ◽

Vol 20 (3) ◽

pp. 359-365 ◽

Cited By ~ 4

Author(s):

A. Sultan

Keyword(s):

General Setting ◽

Continuous Functions ◽

Tychonoff Space ◽

Zero Sets ◽

Rings Of Continuous Functions ◽

Hewitt Realcompactification ◽

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-043-2 ◽

2005 ◽

Vol 57 (6) ◽

pp. 1121-1138 ◽

Author(s):

Michael Barr ◽

R. Raphael ◽

R. G. Woods

Keyword(s):

Locally Compact ◽

Compact Spaces ◽

Open Questions ◽

Tychonoff Spaces ◽

Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.