18. Attaching spaces and cell complexes
In this lecture we introduce a particular cool class of topological spaces, called cell complexes.
The importance of cell complexes (also called CW complexes) in algebraic topology cannot be overstated. This semester we will only scratch the surface: we'll show that many common spaces carry the structure of a cell complex, and in a couple of lectures time we'll define cellular homology, which is a superior version of singular homology tailor-made for cell complexes.
We won't study the main reason they are important until next semester. Roughly speaking, we'll see in Algebraic Topology II that from the point of view of any homology/homotopy functor, every space is a cell complex! More precisely, we'll show that one can "approximate" an arbitrary topological space $X$ by a cell complex $Y$ in such a way that a homology/homotopy functor $H_n$ or $\pi_n$ cannot tell the difference.
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