Related papers: Left-Separating Order Types

The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of…

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…

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 $f:Y_1\rightarrow Y_2$…

Theorem: The topological partition relation omega^{*}-> (Y)^{1}_{2} (a) fails for every space Y with |Y| >= 2^c ; (b) holds for Y discrete if and only if |Y| <= c; (c) holds for certain non-discrete P-spaces Y ; (d) fails for Y= omega cup…

A compact Hausdorff space X is called a CO space, if every closed subset of X is homeomorphic to an open subset of X. Every successor ordinal with its order topology is a CO space. We find an explicit characterization of the class K of CO…

A space $ X $ is said to be set star-Lindel\"{o}f (resp., set strongly star-Lindel\"{o}f) if for each nonempty subset $ A $ of $ X $ and each collection $ \mathcal{U} $ of open sets in $ X $ such that $ \overline{A} \subseteq \bigcup…

In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. \[\text{Characterize the order types…

For a Urysohn space $X$ we define the regular diagonal degree $\overline{\Delta}(X)$ of $X$ to be the minimal infinite cardinal $\kappa$ such that $X$ has a regular $G_\kappa$-diagonal i.e. there is a family $(U_\eta:\eta<\kappa)$ of open…

The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by…

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D,…

A topological space $X$ is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space $X$ has a resolving subset that can be written as an at most countably infinite…

Given a topological property $P$, we say that the space $X$ is $P$-generated if for any subset $A\subset X$ that is not open in $X$ there is a subspace $Y \subset X$ with property $P$ such that $A\cap Y$ is not open in $Y$. (Of course, in…

Let $CH$ be the class of compacta (i.e., compact Hausdorff spaces), with $BS$ the subclass of Boolean spaces. For each ordinal $alpha$ and pair $(K,L)$ of subclasses of $CH$, we define $Lev_{>=alpha}(K,L)$, the class of maps of level at…

We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindel\"of degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\leq 2^{wL(X)\pi\chi(X)\psi(X)}$, and…

We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$.…

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $\alpha(X)$ of compact families of open subsets of a…

Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and…

Let X be a topological space. The closure of \Delta = {(x, x) : x \in X} in X \times X is a symmetric relation on X. We characterise those equivalence relations on an infinite set that arise as the closure of the diagonal with respect to a…

For a smooth spacetime $X$, based on the timelike homotopy classes of its timelike paths, we define a topology on $X$ that refines the Alexandrov topology and always coincides with the manifold topology. The space of timelike or causal…

A space $X$ is of countable type (resp. subcountable type) if every compact subspace $F$ of $X$ is contained in a compact subspace $K$ that is of countable character (resp. countable pseudocharacter) in $X$. In this paper, we mainly show…