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Every finite simplicial complex is weakly homotopy equivalent to a finite space. Therefore there are finite spaces with nontrivial loops; and these are obviously not embedded.

Hahn–Mazurkiewicz Theorem: Suppose $X$ is a nonempty Hausdorff topological space. Then the following are equivalent: there is a surjection $[0,1]\to X$, $X$ is compact, connected, locally connect …

I am looking at the paper Covering homotopy properties of maps between CW complexes or ANRs by Mark Steinberger and James West and a claim is made in the proof of their first main theorem t …

Call $A$ an HM-space if there is a continuous surjection $I\to A$ (where $I$ is the closed interval $[0,1]$). Note that if $A$ is an HM-space, then it is path-connected. Theorem: If $X$ is path-conn …

This question takes place in the category $\mathrm{CGWH}$ of compactly generated weak Hausdorff spaces. Let $\lambda$ be a limit ordinal, and suppose we have a diagram $\Phi: \lambda \to \mathrm{CGWH} …

There is a theorem of Dold to this effect: Dold, Albrecht Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft. (German) Invent. Math. 6 1968 185–189.

Here is the main theorem of "A Vietoris Mapping Theorem for Homotopy" by S. Smale: THEOREM: Let $f:X\to Y$ be proper and onto, where $Y$ and $X$ are $0$-connected separable locally c …

Let $X$ and $Y$ be CW complexes. Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular Approxim …

It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$ to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$ nonhomeomorphic. Is there a standard go-to examp …

No. Take any finite simplicial complex $K$, find an $n$ for which $K$ embeds (piecewise linearly, even!) in $\mathbb{R}^n$. Then for sufficiently small $\varepsilon > 0$, the $\varepsilon$-neighborho …

Suppose I have a pullback square, which I think of as a map from the fibration $q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$ from the mapping cylinder $M$ of $X\t …

Since a retract of a Hausdorff space is closed, such a space must be discrete.

Take the disjoint union of any two nonhomeomorphic spaces with that property as long as they are perfect, e.g., $\mathbb{Q}\coprod(\mathbb{R}-\mathbb{Q})$.

The easiest way I know to say what is going on is to resort to looking at "products" of pairs: $$ (X, A) \times (Y, B) = ( X\times Y , A\times Y \cup X\times B). $$ The point of this notation is that …

I've written a paper (or two) about collection $\mathcal{R}$ of all pointed topological spaces $Y$ satisfying the property $\mathrm{map}_*(X,Y) \sim *$ (for fixed $X$). The interesting fact is that …