Cellular Homology, Compact Subspace of a CW Complex

A Compact Set Intersects Finitely Many Cells

Let T be a compact subset of the cw complex S. Suppose an infinite number of cells in S intersects T. For each cell E_i intersecting T, let p_i be a point in the intersection. Let R_i be the infinite union of all points p₁ p₂ p₃ etc, without p_i. If i ≠ j, E_j intersects R_i in p_j. If i = j, or j is a cell outside our set, apart from T, E_j and R_i are disjoint.

Let F_j be the closure of E_j. Other points from R_i may be present in the boundary of F_j. However, since a finite number of cells contain the boundary, the intersection of F_j and R_i is finite.

Since S is hausdorff, each point is closed, and E_j∩R_i is closed. Thus each R_i is closed in S.

Let O_i be the complement of R_i. Note that O_i forms a cover for S, and for T. Any finite subcover would exclude some point p_n, which contradicts T a compact set. Therefore T intersects a finite number of cells.

Implied Subcomplex

Start with the cells that intersect T. Include the boundaries of these cells, as defined by their respective functions f on unit balls of various dimensions. This brings in finitely many new open cells per cell, each having lower dimension than the open cell that mandated its appearance. Repeat this process until no new open cells appear. For any given cell, the process must terminate, since the maximum dimension of a new cell drops by at least one with each iteration. Do this for all cells in T, and take the union, which could be an infinite union if the cells of T have unbounded dimension.

This collection of cells, call it U, intersects a closed cell in the entire closed cell or nothing. The intersection is always closed, hence U is closed. We want to prove U is a cw complex, that is, a subcomplex of S.

Each characteristic function from a closed ball into the cells of U remains valid. The image of the boundary is covered by finitely many cells of S at lower dimensions, and all these cells are folded into U.

If a subset V is closed in U it is closed in S, and its intersection with any closed cell of S, or of U, remains closed.

This reasoning can be reversed. Let V intersect every closed cell of U in a closed set. Since U is closed, this takes care of all the cells of U. Let E be a cell not contained in U, which means E lies outside of U, since cells are disjoint. If V intersects the boundary of E, it intersects lower dimensional cells (containing the boundary) in closed sets. Combine these (finitely many) intersections together, and the result is still closed. Thus V intersects every closed cell of S in a closed set, and V is closed in S. With V entirely inside U, V remains closed in the subspace topology of U. Therefore U is a cw subcomplex of S.