• 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.

• Korean Journal of Digital Imaging in Medicine
• /
• v.11 no.1
• /
• pp.5-13
• /
• 2009

There are several reasons to take X-ray in case of inpatients. Some of them who cannot ambulate or have any risk if move are taken portable X-ray at their wards. Usually, in this case, many other people-patients unneeded X-ray test, family, hospital workers etc-are indirectly exposed to X-ray by scatter ray. For that reason I try to be aware of free space scatter dose accurately and make the point at issue of portable X-ray better in this study. kVp dose meter is used for efficiency management of portable X-ray equipment. Mobile X-ray equipment, ionization chamber, electrometer, solid water phantom are used for measuring of free space scatter dose. First of all the same surroundings condition is made as taken real portable X-ray, inquired amount of X-ray both chest AP and abdomen AP most frequently examined and measured scatter ray distribution of two tests individually changing distance. In the result of measuring horizontal distribution with condition of chest AP it is found that the mAs is decreased as law of distance reverse square but no showed mAs change according to direction. Vertical distribution showed the mAs slightly higher than horizontal distribution but it isnt found out statistical characteristic. In abdomen AP, compare with chest AP, free space scatter dose is as higher as five-hundred times and horizontal, vertical distribution are quite similar to chest AP in result. In portable X-ray test, in order to reduce the secondary exposure by free space scatter dose first, cut down unnecessary portable order the second, set up the specific area at individual ward for the test the third, when moving to a ward for the X-ray test prepare a portable shielding screen. The last, expose about 2m apart from patients if unable to do above three ways.

• Kyungpook Mathematical Journal
• /
• v.46 no.4
• /
• pp.503-507
• /
• 2006

Let X be a Tychonoff space and ${\sum}(X)$ the set of all the subrings of C(X) that contain $C^*(X)$. For any A(X) in ${\sum}(X)$ suppose $_{{\upsilon}A}X$ is the largest subspace of ${\beta}X$ containing X to which each function in A(X) can be extended continuously. Let us write A(X) ~ B(X) if and only if $_{{\upsilon}A}X=_{{\upsilon}B}X$, thereby defining an equivalence relation on ${\sum}(X)$. We have shown that an A(X) in ${\sum}(X)$ is isomorphic to C(Y ) for some space Y if and only if A(X) is the largest member of its equivalence class if and only if there exists a subspace T of ${\beta}X$ with the property that A(X)={$f{\in}C(X):f^*(p)$ is real for each $p$ in T}, $f^*$ being the unique continuous extension of $f$ in C(X) from ${\beta}X$ to $\mathbb{R}^*$, the one point compactification of $\mathbb{R}$. As a consequence it follows that if X is a realcompact space in which every $C^*$-embedded subset is closed, then C(X) is never isomorphic to any A(X) in ${\sum}(X)$ without being equal to it.

• Journal of the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.129-134
• /
• 2005

Lpt X be a reduced Stein space and L a holomorphic line bundle on X. L is spanned by its global sections and the associated holomorphic map $h_L\;:\;X{\to}P(H^0(X, L)^{\ast})$ is an embedding. Choose any locally convex vector topology ${\tau}\;on\;H^0(X, L)^{\ast}$ stronger than the weak-topology. Here we prove that $h_L(X)$ is sequentially closed in $P(H^0(X, L)^{\ast})$ and arithmetically Cohen -Macaulay. i.e. for all integers $k{\ge}1$ the restriction map ${\rho}_k\;:\;H^0(P(H^0(X, L)^{\ast}),\;O_{P(H^0(X, L)^{\ast})}(k)){\to}H^0(h_L(X),O_{hL_(X)}(k)){\cong}H^0(X, L^{\otimes{k}})$ is surjective.

• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.905-913
• /
• 2022

We characterize metrizability and submetrizability for point-open, open-point and bi-point-open topologies on C(X, Y), where C(X, Y) denotes the set of all continuous functions from space X to Y ; X is a completely regular space and Y is a locally convex space.

• Honam Mathematical Journal
• /
• v.44 no.2
• /
• pp.219-230
• /
• 2022

We examine the bi-rotational hypersurface x = x(u, v, w) with the third Laplace-Beltrami operator in the four dimensional Euclidean space 𝔼⁴. Giving the i-th curvatures of the hypersurface x, we obtain the third Laplace-Beltrami operator of the bi-rotational hypersurface satisfying ∆^IIIx =𝒜x for some 4 × 4 matrix 𝒜.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.4
• /
• pp.603-614
• /
• 2015

For a map $p:X{\rightarrow}A$, we define and study a concept of $G^{\prime}_p$-space for a map, which is a generalized one of a G'-space. Any G'-space is a $G^{\prime}_p$-space, but the converse does not hold. In fact, $CP^2$ is a $G^{\prime}_{\delta}$-space, but not a G'-space. It is shown that X is a $G^{\prime}_p$-space if and only if $G^n(X,p,A)=H^n(X)$ for all n. We also obtain some results about $G^{\prime}_p$-spaces and homology decompositions for spaces. As a corollary, we can obtain a dual result of Haslam's result about G-spaces and Postnikov systems.

• Journal of the Chungcheong Mathematical Society
• /
• v.29 no.2
• /
• pp.287-328
• /
• 2016

In the current work, we define and find the general solution of the decic functional equation g(x + 5y) - 10g(x + 4y) + 45g(x + 3y) - 120g(x + 2y) + 210g(x + y) - 252g(x) + 210g(x - y) - 120g(x - 2y) + 45g(x - 3y) - 10g(x - 4y) + g(x - 5y) = 10!g(y) where 10! = 3628800. We also investigate and establish the generalized Ulam-Hyers stability of this functional equation in Banach spaces, generalized 2-normed spaces and random normed spaces by using direct and fixed point methods.

• Kyungpook Mathematical Journal
• /
• v.55 no.1
• /
• pp.219-224
• /
• 2015

In this paper it is investigated as to when a nonempty invariant closed subset A of a $S^1$-space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and $S^1$ acts on $X{\times}S^1$ by the action $(x,p){\cdot}q=(x,p{\cdot}q)$, where p, $q{\in}S^1$ and $x{\in}X$, then the hyperspace $2^{X{\times}S^1}$ of all nonempty compact subsets of $X{\times}S^1$ has the S-ECIP.

• Journal of the Chungcheong Mathematical Society
• /
• v.1 no.1
• /
• pp.71-76
• /
• 1988

For each continuous convex function ${\phi}$ defined on an open convex subset $A_{\phi}$ of a Banach space X, if we define a positively homogeneous sublinear functional ${\sigma}_x$ on X by ${\sigma}_x(y)=\sup{\lbrace}f(y)\;:\;f{\in}{\partial}{\phi}(x){\rbrace}$, where ${\partial}{\phi}(x)$ is a subdifferential of ${\phi}$ at x, then we get the following characterization theorem of Gateaux differentiability (weak Asplund) sapce. THEOREM. For every ${\phi}$ above, $D_{\phi}={\lbrace}x{\in}A\;:\;\sup_{||u||=1}\;{\sigma}_x(u)+{\sigma}_x(-u)=0{\rbrace}$ contains dense (dense $G_{\delta}$) subset of $A_{\phi}$ if and only if X is a Gateaux differentiability (weak Asplund) space.