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.
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.