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The Sierpinski topology allows us to reformulate open sets an thus any topological property in terms of continuous functions. The Sierpinski topology is the topology on $\{0,1\}$ where the sets $\empt …

We can always get non-principal ultrafilters on sets from non-discrete extremally disconnected spaces. Let $X$ be a topological space. Let $\text{Clo}(X)$ denote the Boolean algebra of clopen subsets …

Yes. The complete atomless Boolean algebra with a countable $\pi$-base is unique up to isomorphism. For a proof, we observe that if $B$ has a countable $\pi$-base $A$, then the Boolean algebra $C$ gen …

There is a way to promote function space topologies so that convergence of nets behaves the way that one would expect it to behave. There are probably other ways of getting a topology for spaces of pa …

I can think of a few proofs of the fact that there is no space $X$ with $[0,1]$ homeomorphic to $X^2$. There are probably more proofs in addition to the proofs in the other answers. Proof 1-1: Suppose …

Recall that an Alexandrov space is a topological space where the intersection of arbitrary collection of open sets is open. Alexandrov duality states that the category of Alexandrov spaces is equivale …

I claim that every autohomeomorphism of $M$ is conjugate to an extension of an autohomeomorphism of the Cantor space. Let $B$ be the Stone space of the Cantor space. Let $\overline{B}$ denote its Bool …

It is much easier to deal with zero dimensional spaces and ultrametrics than connected metrics. $\omega^\omega$ with the standard ultrametric is isometrically isomorphic to its own hyperspace. In part …

Several commenters observed that the topologist's sine curve is a non-path connected subspace of $\mathbb{R}^2$ that can be written as an intersection of countably many simply connected spaces. One ca …

The answer to both of these questions is Yes. And this result can generalize to point-free topology and I consider this result to be more natural in the context of point-free topology. The following o …

Yes. We can always extend a permutation $f:\{0,1\}^n\rightarrow\{0,1\}^n$ to a homeomorphism $F:[0,1]^n\rightarrow[0,1]^n$ whenever $n\geq 3$ or $n=1$. Proposition: Suppose that $X$ is a connected reg …

Yes. For every non-isolated $p\in X$, there is a well-ordered net in $X\setminus\{p\}$ converging to $p$ as long as $X$ is compact and Hausdorff. This result was observed in the 1992 paper Convergent …

This is generally not a frame nor is it a complete lattice. For example, consider $\omega+1$ which is the one-point compactification of the countable discrete space $\omega$. Then the set of all germs …

By collapsing cardinals with forcing, one can derive Brouwer's fixed point theorem for $[0,1]^\kappa$ where $\kappa$ is uncountable from Brouwer's fixed point theorem for $[0,1]^{\aleph_0}$. Suppose t …

Lemma: Let $(U_{\alpha})_{\alpha\in A}$ be a finitely intersecting open cover of a space $X$. Then there is some partition $P$ of $A$ where $(\bigcup_{\alpha\in R}U_\alpha)_{R\in P}$ is a partition of …