scholarly journals Notes on Fréchet spaces

1999 ◽  
Vol 22 (3) ◽  
pp. 659-665 ◽  
Author(s):  
Woo Chorl Hong
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Fréchet Spaces ◽  
Sequential Space ◽  
Countable Space ◽  
Sequential Closure ◽  
Simple Expansion ◽  
First Countable Space

First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

scholarly journals The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

2019 ◽  
Vol 69 (1) ◽  
pp. 185-198
Author(s):  
Fadoua Chigr ◽  
Frédéric Mynard
Keyword(s):  
Space Structures ◽  
Theoretic Approach ◽  
Hausdorff Spaces ◽  
Sequential Space ◽  
Countable Space ◽  
The Stability ◽  
First Countable Space ◽  
Algebraic Proofs ◽  
Non Hausdorff Spaces

AbstractThis article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

scholarly journals A better framework for first countable spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 289
Author(s):  
Gerhard Preuss
Keyword(s):  
Topological Space ◽  
Uniform Space ◽  
Topological Spaces ◽  
Convergence Space ◽  
Convergence Spaces ◽  
Natural Function ◽  
Topological Construct ◽  
Countable Space ◽  
Filter Space ◽  
First Countable Space

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

scholarly journals Intersections of Primary Ideals in Rings of Continuous Functions

1972 ◽  
Vol 24 (3) ◽  
pp. 502-519 ◽  
Author(s):  
R. Douglas Williams
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Necessary And Sufficient ◽  
Primary Ideals

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

Pointwise Sequentially Closed Ideals in C*(X)

1973 ◽  
Vol 16 (1) ◽  
pp. 115-117 ◽  
Author(s):  
Richard G. Wilson
Keyword(s):  
Pointwise Convergence ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Maximal Ideals ◽  
Sequential Closure ◽  
Convergence Of Sequences ◽  
Necessary And Sufficient

The purpose of this paper is to determine the conditions under which the maximal ideals of the ring C*(X)—the bounded real-valued continuous functions on a completely regular Hausdorff space X—are closed under pointwise convergence of sequences. Whereas the maximal ideals of C*(X) are closed under pointwise convergence of nets if and only if X is compact, it is shown that a necessary and sufficient condition for their pointwise sequential closure is that X be pseudocompact (i.e. that all real-valued continuous functions of X be bounded).

Ideal convergence of nets of functions with values in uniform spaces

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6281-6292
Author(s):  
Athanasios Megaritis
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Sets ◽  
Uniform Spaces ◽  
Ideal Convergence ◽  
Necessary And Sufficient

We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.

scholarly journals The partially pre-ordered set of compactifications of Cp(X, Y)

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. Dorantes-Aldama ◽  
R. Rojas-Hernández ◽  
Á. Tamariz-Mascarúa
Keyword(s):  
Ordered Set ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Countable Space ◽  
Generalized Ordered Space ◽  
Ordered Space ◽  
Countable Character ◽  
Corson Compact ◽  
First Countable Space ◽  
Isolated Point

AbstractIn the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

Eberlein compacts and spaces of continuous functions

1979 ◽  
Vol 85 (2) ◽  
pp. 305-313
Author(s):  
Richard J. Hunter ◽  
J. W. Lloyd
Keyword(s):  
Topological Space ◽  
Weak Topology ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Locally Convex Spaces ◽  
Weakly Compact ◽  
Spaces Of Continuous Functions ◽  
Eberlein Compact ◽  
Necessary And Sufficient ◽  
Locally Convex

AbstractLet X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.

scholarly journals Note on pointwise contractive projections

1993 ◽  
Vol 16 (4) ◽  
pp. 817-818
Author(s):  
L. M. Sanchez Ruiz ◽  
J. R. Ferrer Villanueva
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Compact Open Topology ◽  
Completely Regular ◽  
Contractive Projection ◽  
Necessary And Sufficient ◽  
Contractive Projections

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

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