• Honam Mathematical Journal
• /
• v.39 no.4
• /
• pp.655-663
• /
• 2017

In this paper we define $AP_c$ and $AP_{cc}$ spaces which are stronger than the property of approximation by points(AP). We investigate operations on their subspaces and study function theorems on $AP_c$ and $AP_{cc}$ spaces. Using those results, we prove that every continuous image of a countably compact Hausdorff space with AP is AP. Finally, we prove a theorem that every compact ${\kappa}$-WAP space is ${\kappa}$-pseudoradial, and prove a theorem that the product of a compact ${\kappa}$-radial space and a compact ${\kappa}$-WAP space is a ${\kappa}$-WAP space.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.1105-1109
• /
• 2011

We establish a result concerning simultaneous approximation and interpolation from certain uniformly dense subsets of the space of vector-valued continuous functions vanishing at infinity on locally compact Hausdorff spaces.

• Journal of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.155-167
• /
• 1996

Let X be a compact Hausdorff space, C(X) denote the set of all continuous real valued functions on X and $\ell \in N$ be any natural number.

• Journal of the Korean Mathematical Society
• /
• v.45 no.1
• /
• pp.197-204
• /
• 2008

We study diameter preserving linear maps from C(X) into C(Y) where X and Y are compact Hausdorff spaces. By using the extreme points of $C(X)^*\;and\;C(Y)^*$ and a linear condition on them, we obtain that there exists a diameter preserving linear map from C(X) into C(Y) if and only if X is homeomorphic to a subspace of Y. We also consider the case when X and Y are noncompact but locally compact spaces.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.457-467
• /
• 2011

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

• The Mathematical Education
• /
• v.16 no.2
• /
• pp.22-26
• /
• 1978

It is shown that a map having an extension to an open map between the Alex-androff base compactifications of its domain and range has a unique such extension. J.S. Wasileski has introduced the Alexandroff base compactifications of Hausdorff spaces endowed with Alexandroff bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by ABC. We prove that the Alexandroff base compactification on objects can be extended to a functor on ABC and that the compact objects give an epireflective subcategory of ABC. For each topological space X there exists a completely regular space $\alpha$X and a surjective continuous function $\alpha$$_{x}$ : Xlongrightarrow$\alpha$X such that for each completely regular space Z and g$\in$C (X, Z) there exists a unique g$\in$C($\alpha$X, 2) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\alpha$$_{x}$, $\alpha$X) is called a completely regularization of X. Let TOP be the category of topological spaces and continuous functions and let CREG be the category of completely regular spaces and continuous functions. The functor $\alpha$ : TOPlongrightarrowCREG is a completely regular reflection functor. For each topological space X there exists a compact Hausdorff space $\beta$X and a dense continuous function $\beta$x : Xlongrightarrow$\beta$X such that for each compact Hausdorff space K and g$\in$C (X, K) there exists a uniqueg$\in$C($\beta$X, K) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\beta$$_{x}$, $\beta$X) is called a Stone-Cech compactification of X. Let COMPT$_2$ be the category of compact Hausdorff spaces and continuous functions. The functor $\beta$ : TOPlongrightarrowCOMPT$_2$ is a compact reflection functor. For each topological space X there exists a realcompact space (equation omitted) and a dense continuous function (equation omitted) such that for each realcompact space Z and g$\in$C(X, 2) there exists a unique g$\in$C (equation omitted) with g=g$^{\circ}$(equation omitted). Such a pair (equation omitted) is called a Hewitt's realcompactification of X. Let RCOM be the category of realcompact spaces and continuous functions. The functor (equation omitted) : TOPlongrightarrowRCOM is a realcompact refection functor. In [2], D. Harris established the existence of a category of spaces and maps on which the Wallman compactification is an epirefiective functor. H. L. Bentley and S. A. Naimpally [1] generalized the result of Harris concerning the functorial properties of the Wallman compactification of a T$_1$-space. J. S. Wasileski [5] constructed a new compactification called Alexandroff base compactification. In order to fix our notations and for the sake of convenience. we begin with recalling reflection and Alexandroff base compactification.

• Honam Mathematical Journal
• /
• v.27 no.4
• /
• pp.631-639
• /
• 2005

For compact Hausdorff spaces X and Y let C(X) and C(Y) denote the complex Banach spaces of complex valued continuous functions on X and Y, respectively. Also let $T:C(X){\rightarrow}C(Y)$ be a linear isomerty. Necessary and sufficient conditions are given such that range of T has finite codimension.

• Communications of the Korean Mathematical Society
• /
• v.22 no.4
• /
• pp.569-573
• /
• 2007

In this note, we construct an example of a Hausdorff K-starcompact (hence, $1\frac{1}{2}$-star-compact) space X having a regular closed $G_{\delta}-subset$ which is not $1\frac{1}{2}$-starcompact (hence, not K-starcompact).

• Journal of the Chungcheong Mathematical Society
• /
• v.36 no.4
• /
• pp.279-288
• /
• 2023

We obtain a Chandrabhan type fixed point theorem for a multimap having a non-compact domain and a weakly closed graph, and taking convex values only on a quasi-convex subset of Hausdorff locally convex topological vector space. We introduce the definition of Chandrabhan-set and find a sufficient condition for every countably condensing multimap to have a relatively compact Chandrabhan-set. Finally, we establish a new version of Sadovskii fixed point theorem for multimaps.

• Korean Journal of Mathematics
• /
• v.29 no.1
• /
• pp.25-40
• /
• 2021

In this article we introduce tribonacci sequence spaces c0(T) and c(T) derived by the domain of a newly defined regular tribonacci matrix T. We give some topological properties, inclusion relations, obtain the Schauder basis and determine ��-, ��- and ��- duals of the spaces c0(T) and c(T). We characterize certain matrix classes (c0(T), Y) and (c(T), Y), where Y is any of the spaces c0, c or ℓ∞. Finally, using Hausdorff measure of non-compactness we characterize certain class of compact operators on the space c0(T).