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No, the closure of the image of $f$ in $Y$ is never the Stone-Čech compactification of $X$ unless $X$ is empty. In particular, consider the element $a\in Y$ which is $1$ on every coordinate. Note th …

Any countable Hausdorff space $Q$ is totally path-disconnected. Indeed, if $f:[0,1]\to Q$ is continuous, then its image $X$ is a countable connected compact Hausdorff space. By Urysohn's lemma, then …

The monoid $\mathcal M_{\rm fin}$ is in fact cancellative. To prove this, we start with a Lemma. Let us write $h(T,Y)$ for the cardinality of the set of continuous maps $T\to Y$ and $i(T,Y)$ for t …

This follows from Alexander duality, and works with $\mathbb{R}^2$ replaced by $\mathbb{R}^d$ for any $d$. In more detail, let us compactify $\mathbb{R}^d$ to $S^d$ and consider $F'=S^d\setminus\bigc …

Any map $[0,1]\to[0,1]$ is a quotient map onto its image, and the image must be either a point or a closed interval. So a partition comes from such a map iff the quotient of $[0,1]$ by the equivalenc …

Yes, you can always connect $f_0$ and $f_1$ by a finite sequence of good homotopies supported in any given open cover (and in fact, this sequence of good homotopies can be chosen to be homotopic to $f …

No, there is not, even if you do not require $f$ to be injective. To see this, let $F$ be any ultrafilter on $\kappa$, and consider the following space $X_F$. The underlying set of $X_F$ is $\kappa\ …

A pseudo-arc is an example of a compact connected subset of the plane that is totally path-disconnected.

No. In fact, every compact Hausdorff space $K$ embeds in a locally convex topological vector space. Namely, take the dual $C(K)'$ of the space of continuous functions on $K$ with the weak* topology, …

Let $B$ and $C$ be convex regular open sets such that $B\cdot C\neq 0$, and write $D=B-C$ and $E=C-B$. It suffices to show that the union $D\cup E$ is regular. Indeed, if it is, then $B\oplus C=D\cu …

Let $Z$ be the Sierpinski 2-point space. Then the underlyying set of $C(X,Z)$ is naturally identified with the collection of open sets of $X$, and the specialization order from the compact-open topol …

This phenomenon is very typical in compact spaces whose construction involves taking an uncountable product. For instance, consider $X=\{0,1\}^{\mathcal{P}(\mathbb{N})}$ with the product topology. C …

Here's a simple example. Let $X=\mathbb{N}^2\cup\{\infty\}$ topologized such every subset of $\mathbb{N}^2$ is open and the neighborhood filter at $\infty$ is generated by the sets $\mathbb{N}\times[ …

It is very easy to see that CG-ification preserves weak homotopy type: it is a right adjoint, so for any CG space $K$, the maps $K\to X_{CG}$ are the same as the maps $K\to X$. Letting $K$ be $S^n$ a …

First, note that $f:X\to Y$ must be locally injective. Now choose a splitting of $f$ and consider $Y$ as a subspace of $X$ via this splitting. For any $y\in Y$, there then some neighborhood $U\subse …