arXiv subjects

M. R. Koushesh

Publications and source records attributed to M. R. Koushesh.

24 records · Page 2Linked to original sources

arXiv2012

Compactification-like extensions

Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness then $P$-extensions are called ompactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like $P$-extensions, where $P$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like $P$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like $P$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like $P$-extensions of a space among all its Tychonoff $P$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like $P$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $βX\backslash X$. We conclude with some applications, including an answer to an old question of S. Mrówka and J.H. Tsai: For what pairs of topological properties $P$ and $Q$ is it true that every locally-$P$ space with $Q$ has a one-point extension with both $P$ and $Q$?

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arXiv2012

On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space $X$ is dense in a Tychonoff space $Y$, then $Y$ is called a Tychonoff extension of $X$. Two Tychonoff extensions $Y_1$ and $Y_2$ of $X$ are said to be equivalent, if there exists a homeomorphism $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of $X$. We identify those extensions of $X$ which belong to the same equivalence classes. For two Tychonoff extensions $Y_1$ and $Y_2$ of $X$, we write $Y_2\leq Y_1$, if there exists a continuous function $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This is a partial order on the set of all Tychonoff extensions of $X$. If a Tychonoff extension $Y$ of $X$ is such that $Y\backslash X$ is a singleton, then $Y$ is called a one-point extension of $X$. Let $T(X)$ denote the set of all one-point extensions of $X$. We study the order structure of the partially ordered set $(T(X),\leq)$. For a locally compact space $X$, we define an order-anti-isomorphism from $T(X)$ onto the set of all non-empty closed subsets of $βX\backslash X$. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of $X$, the set of all one-point Lindelof extensions of $X$, the set of all one-point pseudocompact extensions of $X$, and the set of all one-point Cech-complete extensions of $X$, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of $βX\backslash X$. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of $C^*(X)$ and the topology of subspaces of $βX\backslash X$.

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arXiv2012

On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space $X$, let ${\mathcal E}(X)$ be the set of all one-point metrizable extensions of $X$, and when $X$ is locally compact, let ${\mathcal E}_K(X)$ denote the set of all locally compact elements of ${\mathcal E}(X)$ and $λ: {\mathcal E}(X)\rightarrow{\mathcal Z}(βX\backslash X)$ be the order-anti-isomorphism (onto its image) defined in: [HJW] M. Henriksen, L. Janos and R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolinae 46 (2005), 105-123. By definition $λ(Y)= \bigcap_{n<ω}cl_{βX}(U_n\cap X)\backslash X$, where $Y=X\cup\{p\}\in{\mathcal E}(X)$ and $\{U_n\}_{n<ω}$ is an open base at $p$ in $Y$. Answering the question of [HJW], we characterize the elements of the image of $λ$ as exactly those non-empty zero-sets of $βX$ which miss $X$, and the elements of the image of ${\mathcal E}_K(X)$ under $λ$, as those which are moreover clopen in $βX\backslash X$. We then study the relation between ${\mathcal E}(X)$ and ${\mathcal E}_K(X)$ and their order structures, and introduce a subset ${\mathcal E}_S(X)$ of ${\mathcal E}(X)$. We conclude with some theorems on the cardinality of the sets ${\mathcal E}(X)$ and ${\mathcal E}_K(X)$, and some open questions.

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arXiv2012

One-point extensions of locally compact paracompact spaces

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} if $Y\backslash X$ is a singleton. An extension $Y$ of $X$ is called {\em first-countable} if $Y$ is first-countable at points of $Y\backslash X$. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} if it has ${\mathcal P}$. In this article, for a given locally compact paracompact space $X$, we consider the two classes of one-point Čech-complete ${\mathcal P}$-extensions of $X$ and one-point first-countable locally-${\mathcal P}$ extensions of $X$, and we study their order-structures, by relating them to the topology of a certain subspace of the outgrowth $βX\backslash X$. Here ${\mathcal P}$ is subject to some requirements and include $σ$-compactness and the Lindelöf property as special cases.

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arXiv2012

The partially ordered set of one-point extensions

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} of $X$ if $Y\backslash X$ is a singleton. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} of $X$ if it has ${\mathcal P}$. One-point ${\mathcal P}$-extensions comprise the subject matter of this article. Here ${\mathcal P}$ is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space $X$ (partially ordered by $\leq$) and the set of compact non-empty subsets of its outgrowth $βX\backslash X$ (partially ordered by $\subseteq$). This enables us to study the order-structure of various sets of one-point extensions of the space $X$ by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces $X$ denote by ${\mathscr U}(X)$ the set of all zero-sets of $βX$ which miss $X$. \noindent{\bf Conjecture.} {\em For locally compact spaces $X$ and $Y$ the partially ordered sets $({\mathscr U}(X),\subseteq)$ and $({\mathscr U}(Y),\subseteq)$ are order-isomorphic if and only if the spaces ${\em cl}_{βX}(βX\backslash\upsilon X)$ and ${\em cl}_{βY}(βY\backslash\upsilon Y)$ are homeomorphic.}

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arXiv2012

A pseudocompactification

For a locally pseudocompact space $X$ let [ζX=X\cup cl_{βX}(βX\backslash\upsilon X).] It is proved that $ζX$ is the largest (with respect to the standard partial order $\leq$) among all pseudocompactifications of $X$ which have compact remainder. Other characterizations of $ζX$ are also given.

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