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Is there a ZFC example of a Tychonoff space $X$ such that: $X$ is separable. $X$ has countable tightness (that is, a subset of $X$ is closed if and only if it contains the closure of each one of its …

Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non …

An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first const …

A space $X$ is said to be sequential if whenever $A \subset X$ is not closed then $A$ contains a sequence converging to a point outside of $A$. A space $X$ is said to have countable tightness if for …

One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is the inter …

All spaces are assumed to be Hausdorff. Recall that a cellular family in the space $X$ is a family of pairwise disjoint non-empty open subspaces of $X$. The cellularity of $X$ ($c(X)$) is defined as t …

Background/Motivation A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) converg …

The space $\omega^*$, the remainder of the Cech-Stone compactification of the integers, fails to have all convergence-type properties known to me. Sequentiality. (As a matter of fact $\omega^*$ does …

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces? Recall that a sp …

Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable. See: Dow, Alan; Tall, Franklin D., Hereditarily nor …

A non-empty topological space without isolated points is called maximal if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple c …

Jensen proved that under $\Diamond$ there is a homogeneous Suslin continuum, so the square of a ccc homogeneous space can fail to be ccc. What about ccc topological groups? Is there a ccc top …

In a recent MO post it was noted that Uspenskij's old example of a Tychonoff ccc space with a $G_\delta$ diagonal and arbitrarily large cardinality is not normal. See: How could I see quickly that th …

Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical applicat …

Given a topological space $X$, let $X_\delta$ be the topology on $X$ generated by the $G_\delta$ subsets of $X$. Let $c(X)$ be the cellularity of $X$, that is, the supremum of cardinalities of familie …