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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 …

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

Let $\mathcal{U} = \tau \cup \tau^\star$, and let $\tau'$ be the unique minimal topology on $X$ containing $\mathcal{U}$. Since $\tau$ and $\tau^\star$ are topologies, they are closed under finite i …

Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it. Let's say a cone map is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) …

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's say $C$ divides $S^2$ into two disks, $D$ and $E$. Then, choosing a homeomorphism (fixing $C$) $h:D\to E$, we get an involution $t: S^2\to S^2$, which has degree $-1$. On the other hand, sinc …

What should I call a property (P) of (open) subspaces of a space $X$ such that: If $U$ satisfies (P), then so does every open subset $V\subset U$ If {$U_i$} is a pairwise disjoint collection of se …

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 …

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 …

You can show the evaluation map is a weak fibration (I think this is the term: I mean a map homotopy equivalent in the category of spaces over $X\times X$ to a fibration -- namely $X^I\to X\times X$) …

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

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 …

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 …

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.