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(a) Let $X$ be an arbitrary topological space, and $U,V$ open subsets with the same boundary $B$, and $W=U\cap V$. Then the boundary of $W$ is contained in $B$ (immediate). (b) Next, if $U,W$ are ope …

Pick $X$ such that there are path-components $Y \neq Z$, and $y \in Y$, $z\in Z$ with the conditions: $y$ belongs to the interior of $Y$, $z$ belongs to the interior of $Z$, and $Y$ is not a singleton …

Your topology is by definition the profinite topology of $\mathbf{Z}$, restricted to the nonnegative numbers $\mathbf{N}$. Hence it's a nonempty countable metrizable space. By a classical theorem of S …

It's not connected. Let $u=(u_n)$ be a sequence. For $\ell\in [0,1]$, define $$V_{u,\ell}=\{(v_n):\forall n\in \omega:|v_n-\ell|<\max(2^{-n},2|u_n-\ell|)\}.$$ Then $u\in V_{u,\ell}$ and $V_{u,\ell}$ …

Let $X$ be a countable compact Hausdorff space consisting of a unique accumulation point $0$ with discrete complement $X'$. Let $j$ be a bijection $\mathbf{N}\to X'\times\mathbf{N}$. For $n\in\mathb …

Yes. If you take two distinct points outside $A$, there are two clopen subsets disjoint of $A$ and separating them, hence at the quotient these map to clopen subsets, separating them. If you take a po …

There are compact groups that are connected, but not locally connected at any point. For instance, solenoids $(\mathbf{R}\times\mathbf{Z}_p)/\langle (1,1)\rangle$. (It is locally homeomorphic to $\mat …

Assume the contrary. If $x\in C$, then it follows from the Schoenflies theorem $x$ is the limit of points $x_i$ in $A$ and $y_i$ in $B$. If we assume $x\in C\smallsetminus (-C)$, we can assume in add …

The definition is a bit misleading as syndetic subgroups are not assumed to be normal, but you require them to be normal in defining "syndetically separated". So let me change the definition, removing …

I'll assume $E$ Hausdorff otherwise there are trivial counterexamples. Then this fact holds. We can suppose $A$ is dense. Then the map $\Phi:E\to 2^{2^A}$ mapping $x$ to the set of $P\subset A$ such t …

$\DeclareMathOperator{\R}{\mathbf{R}}\DeclareMathOperator{\Z}{\mathbf{Z}}$The answer is no, even in the circle (and hence in the plane). As coordinates, write the circle as the 1-point compactificatio …

It's not true. Here's an easy way to get $[0,1]$ as a limit. Consider $C_0$ as the set of all sequences of $0,1$: each such sequence defines a binary expansion of some element in $[0,1]$. Let $C_n$ b …

The question is only reasonable if one assumes the groups to be Hausdorff, and also to restrict to cardinal $\le 2^c$, since any Hausdorff separable space has cardinal $\le 2^c$. In sharp contrast to …

I'm not properly answering since you're asking for a reference and I don't know any; however here's a hopefully reasonably concise proof. First implication: (1) Let $X,Y$ be compact Hausdorff top …

Here are: 1) a metrizable example. Namely, consider a complete graph on $\alpha\ge\mathfrak{c}$. Its 1-skeleton, endowed with the geodesic metric with edges of length one, has cardinal $\alpha$ and …