# Questions tagged [cw-complexes]

For questions about CW complexes (topological spaces which are built up using balls of varying dimensions known as cells).

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### Compact subset of CW complex

Let $X$ be a CW complex. Prove that a set $A\subseteq X$ is compact if and only if $A$ is closed and has a nonempty intersection with finitely many open cells of $X$ only. What I know: compact means ...
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### CW complex for Möbius strip

I was asked to find a CW complex for the Möbius strip with one 0-cell, two 1-cells, and one 2-cell. I can find a CW complex for a Möbius strip with more cells (two 0-cells, three 1-cells and a single ...
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### Universal covering space of $\mathbb{R^3}\setminus S^1$

I am having a problems with describing the universal covering space of $\mathbb{R^3}\setminus S^1$. Namely, I need to find a CW-space which it is homotopy equivalent to. Well, I don't know how to ...
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### If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals its closure

If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ I'm trying to prove the above which is a statement in Introduction to ...
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### CW complexes are compactly generated

I am looking for verification and/or criticism of a proof I have constructed. The problem is to show that every CW complex is compactly generated. A topological space $X$ is $compactly$ $generated$ if ...
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### CW complex, but with holes in cells

A CW complex is a specific kind of a partition of a topological space by open cells where each open n-cell is homeomorphic to an open n-ball. Consider a modified concept where each open n-cell is ...
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### CW complex infinite-dimensional

Lemma: If $X$ is a CW complex then: a) $H_{k}(X^{n},X^{n-1})$ is zero for $k\neq n$ and is free abelian for $k=n$ with a basis in one-to-one correspondence with the n-cell of $X$. b) $H_{k}(X^{n})=0$...
### If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ [duplicate]
If $(X, \Gamma)$ is a cell complex and $e \in \Gamma$, then the image of a characteristic map for $e$ equals $\bar{e}$ I'm trying to prove the above which is a statement in Introduction to ...