Abstract:
A topological space (,) is said to be strongly compact if every preopen cover of (,) admits a finite subcover. In this paper, we introduce a new class of sets called 𝒩-preopen sets which is weaker than both open sets and 𝒩-open sets. Where a subset is said to be 𝒩-preopen if for each ∈ there exists a preopen set containing such that ？ is a finiteset. We investigate some properties of the sets. Moreover, we obtain new characterizations and preserving theorems of strongly compact spaces.

Abstract:
Mashhour et al. [1] introduced the notions of P1-paracompactness and P2-paracompactness of topological spaces in terms of preopen sets. In this paper, we introduce and investigate a weaker form of paracompactness which is called P3-paracompact. We obtain various characterizations, properties, examples, and counterexamples concerning it and its relationships with other types of spaces. In particular, we show that if a space (X,T) is quasi-submaximal, then (X,T) is paracompact if it is P3-paracompact.

Abstract:
Let n and m be infinite cardinals with n ￠ ‰ ¤m and n be a regular cardinal. We prove certain implications of [n,m]-strongly paracompact, [n,m]-paracompact and [n,m]-metacompact spaces. Let X be [n, ￠ ]-compact and Y be a [n,m]-paracompact (resp. [n, ￠ ]-paracompact), Pn-space (resp. wPn-space). If m= ￠ ‘k Keywords strongly paracompact and metacompact spaces.

Abstract:
We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: \proclaim{Theorem} Suppose X is a paracompact space. There is a CW complex K such that {a.} K is an absolute extensor of X up to homotopy, {b.} If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. proclaim The proof is based on the following simple result (see 1.6). \proclaim{Theorem} Suppose X be a paracompact space and $f:A\to Y$ is a map from a closed subset A of X to a space Y. f extends over X if Y is the union of a family $\{Y_s\}_{s\in S}$ of its subspaces with the following properties: {a.} Each $Y_s$ is an absolute extensor of X, {b.} For any two elements s and t of S there is $u\in S$ such that $Y_s\cup Y_t\subset Y_u$, {c.} $A=\bigcup\limits_{s\in S} \int_A(f^{-1}(Y_s))$. proclaim That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.

Abstract:
We let (X, t ) be a topological space and (X, t*) its semiregularization. Then a topological property P is semiregular provided that t has property P if and only if t* has the same property. In this work, we study semiregular property on some generalizations of compact and paracompact spaces; namely, paracompact, nearly paracompact, weakly compact and weakly paracompact spaces.

Abstract:
Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq \mathcal{C}_{p'}$ and (ii) each $\mathcal{C}_p$ is locally finite. Then $X$ is \emph{$P$-paracompact} if every open cover has a $P$-locally finite open refinement. Further, $X$ is \emph{$P$-metrizable} if it has a $(P \times \mathbb{N})$-locally finite base. This work provides the first detailed study of $P$-paracompact and $P$-metrizable spaces, particularly in the case when $P$ is a $\mathcal{K}(M)$, the set of all compact subsets of a separable metrizable space $M$ ordered by set inclusion.

Abstract:
Let $\Omega$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet boundary condition, and the heat semigroup generated by $L$ have the Gaussian property $(G_{\mathrm{diam}(\Omega)})$ with the regularity of their kernels measured by $\mu\in(0,1]$, where $\mathrm{diam}(\Omega)$ denotes the diameter of $\Omega$. Let $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_{\Phi}\in(n/(n+\mu),1]$. In this paper, the authors introduce the Orlicz-Hardy space $H_{\Phi,\,r}(\Omega)$ by restricting arbitrary elements of the Orlicz-Hardy space $H_{\Phi}(\mathbb{R}^n)$ to $\boz$ and establish its atomic decomposition by means of the Lusin area function associated with $\{e^{-tL}\}_{t\ge0}$. Applying this, the authors obtain two equivalent characterizations of $H_{\Phi,\,r}(\boz)$ in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$.

Abstract:
Characterizations of paracompact finite $C$-spaces via continuous selections are given. We apply these results to obtain some properties of finite $C$-spaces. Factorization theorems and a completion theorem for finite $C$- spaces are also established.

Abstract:
Shape theory works nice for (Hausdorff) paracompact spaces, but for spaces with no separation axioms, it seems to be quite poor. However, for finite and locally finite spaces their weak homotopy type is rather rich, and is equivalent to the weak homotopy type of finite and locally finite polynedra, respectively. In the paper there is proposed a variant of shape theory called quasi-shape, which suits both paracompact and locally finite spaces, i.e. the quas-shape is isomorphic to the weak homotopy type for locally finite spaces, and is \natural-equivalent to the ordinary shape in the case of paracompact spaces.

Abstract:
A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is $\pi$-metrizable if and only if $X$ has a $\sigma$-hereditarily closure-preserving $\pi$-base; (2) $X$ is $\pi$-metrizable if and only if $X$ is almost $\sigma$-paracompact and locally $\pi$-metrizable; (3) Open and closed maps preserve $\pi$-metrizability; (4) $\pi$-metrizability satisfies hereditarily closure-preserving regular closed sum theorems. Moreover, we define the notions of second-countable $\pi$-metrizable and strongly $\pi$-metrizable spaces, and study some related questions. Some questions about strongly $\pi$-metrizability are posed.