I'm reading a paper which has this sentence: if $X$ is a CW complex then $|X|$ denotes its underlying topological space. I'm confused about what this underlying topological space is.

  • $\begingroup$ Can you provide more context? As it stands, it's impossible to answer your question. $\endgroup$ – Math1000 Nov 30 '19 at 22:20
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    $\begingroup$ Strictly speaking a CW complex is a space with an associated CW structure. It is likely your text is making clear that a CW complex is more than just a space and different CW complexes can have the same underlying space. $\endgroup$ – Connor Malin Nov 30 '19 at 22:45
  • $\begingroup$ I think the answers you're getting are the ones you should read. But... I encountered this in a paper by Milnor on Whitehead Torsion. So on the off chance that this is his paper, after reading the rest of the paper I'm pretty sure he meant that what he's saying works for simplicial complexes $X$ with geometric realization $|X|$, too, and in the case of a CW complex you can basically ignore the confusion. $\endgroup$ – kamills Dec 1 '19 at 0:04

When talking about a CW complex, the complex itself has more structure than a simple topological space. Each point is partitioned into a particular cell of the complex. However, CW complexes are mostly important because they allow us to manipulate topological spaces in an efficient manner. Some good examples of CW complexes are the surfaces of a cube, or a tetrahedron, or an octahedron, but all of these have the same underlying topological structure, homeomorphic to the surface of a sphere.


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