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Not always. One way to see this is to find $F$ such that $\overline{F^\circ} - F^\circ$ has uncountably many connected components. Since a continuous image of $\mathbb{R} \times \mathbb{Z}$ has only …

Using a relatively recent theorem, we can get a whole family of examples. Suppose $A$ and $B$ are two closed subsets of the real line satisfying the following properties: zero-dimensional $\sigma$- …

There are plenty of topologies on a countable set for which all convergent sequences are eventually constant. The most constructive example I know is the Arens-Fort space, given as example 26 in Stee …

[Edit: Originally my second point below stated incorrectly that if $\tau_1$ and $\tau_2$ are as described, then every topology in between is metrizable. This is not what the theorem in my paper says, …

Let $\tau_1$ be the usual topology on the real line, and let $\tau_2$ be the finer topology obtained by breaking off the positive reals to form a clopen set: that is, $\tau_2$ has a subbasis consistin …

Suppose we have such a topology on $\mathbb{N}$. Let $\{A_\alpha : \alpha < \mathfrak{c}\}$ be an uncountable almost disjoint family of subsets of $\mathbb{N}$. For each $\alpha$, let $$B_\alpha = \bi …

In the comments, Lajos Soukup pointed out the following paper, which seems to contain lots of relevant information: D. Shakhmatov, M. Tkachenko, and R. G. Wilson, "Transversal and $T_1$-independen …

No. [I'm assuming you means to write "let $E_C$ be the connected component of $X \setminus C$ that contains $x^*$." Otherwise I don't see how your question makes sense. Let me know if I'm guessing wr …

Not always. The first uncountable ordinal $\omega_1$, when given the usual order topology, provides a counterexample. That this space is locally compact is pretty well known. I claim that no (closed …

Yes, every non-Hausdorff topology is contained in a maximal non-Hausdorff topology. To see this, let's start with a different question: What do the maximal non-Hausdorff topologies on an infinite set …

For any cardinal $\kappa$ at least the size of the continuum, the "really long line" of length $\kappa$ is an example. This space, let's call it $L_\kappa$, is defined as follows. Begin with the ordi …

Yes, there are such spaces. One example is referenced by Ramiro de la Vega in his answer to this related question. As Joel points out in the comments, the property you describe, namely "$X$ has an an …

The sort of space you describe is usually called strongly $\kappa$-homogeneous. If you google that phrase you will find some interesting results about these kinds of spaces (mostly concerning how this …

A topological space is called irresolvable if it is not the disjoint union of two dense subsets. So you are asking whether there is a connected, $T_2$, irresolvable space with more than one point. The …

No, not always. Let $\tau$ denote the initial segment topology on $\mathbb N$: a subset of $\mathbb N$ is considered open if and only if it is an initial segment of $\mathbb N$. This topology is ultra …