Search Results

Write $F(X)=[X]^2$. Then $F$ is an endofunctor on the category of Hausdorff spaces and inclusions of clopen subspaces, and it preserves filtered colimits. Given any Hausdorff space $X$ with an inclu …

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 …

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\ …

Say a space $X$ is totally retracting if every nonempty subset of $X$ is a retract. Then I claim that $X$ is totally retracting iff its $T_0$ quotient is a disjoint union of spaces $X_i$ such that ea …

The infinite symmetric product of a pointed metric space $(X,e)$ is metrizable iff the basepoint $e$ is isolated. First, if $e$ is isolated and $X=Y\coprod \{e\}$, then $SP(X,e)=\coprod SP^n(Y)$ and …

While johndoe gave the right answer to the question I believe you meant to ask, the question that you did ask is slightly different and also has a negative answer. Specifically, johndoe's answer addr …

Let $X$ be an uncountable set with the cofinite topology. Then I claim there is no coarser minimal $T_0$ topology. Indeed, by Joseph Van Name's answer to the previous question, any minimal $T_0$ top …

Let $X$ be any compactly generated connected Hausdorff space that is not locally compact and let $Y=X\cup\{\infty\}$ be its one-point compactification. Then $Y$ is compact, connected, and not Hausdor …

Here is what I think is a more conceptual explanation of what is going on in Jeremy Rickard's answer. There are two different (and dual) ways to "generate" a topology: you can specify that some sets …

Yes. If you restrict $h$ to any compact subset $E$, then $h$ gives a homeomorphism from $E$ to $h(E)$, because a subset of $E$ (or $h(E)$) is closed iff it is compact, so $h$ and its inverse both pre …

First, note that a minimal totally separated space is the same thing as a Stone space. Clearly Stone spaces are minimal totally separated (any coarser topology cannot even be Hausdorff); conversely s …

You can define a (pseudo)metric on a quotient of a metric space. Let $X$ be a metric space with metric $d$ and an equivalence relation $\sim$. Say that a chain between two points $x,y\in X$ is a seq …

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

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[ …

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 …