Cellular Homology, Skeleton
Skeleton
Let S be a cw complex.
The collection of cells in S with dimension p or less is the p skeleton of S.
This is sometimes denoted Sp.
Skeleton is Closed
Take the closure of Sp by taking the union of the closure of each cell of dimension p or less.
The boundary of each cell is contained in cells of lower dimension, and is already in Sp.
Therefore Sp is closed in S.
Let y be any point in S0.
Let T be the points of S0 without y.
Intersect T with a closed cell E.
The boundary is covered by finitely meny cells, hence finitely many points live in the intersection.
Since S is hausdorff, each point is closed.
The intersection consists of finitely many points, and it too is closed.
This holds for each cell, hence T is closed.
Since y was arbitrary, every point is open, and S0 has the discrete topology.
Sp is a Subcomplex
The cells and characteristic functions of Sp are inherited from S.
Each boundary is covered by finitely many cells at lower dimensions.
We only need verify the topology.
Let T ⊆Sp intersect each closed cell in a closed set.
Let E be a cell in Sp.
Since Sp is closed, the closure of E is also in Sp.
We already know T intersect the closure of E is closed.
And this lives in a closed set in S, hence it is closed in S.
Let E be a cell outside of Sp.
Its boundary is covered by a finite number of cells from Sp.
Each intersection is closed, and their finite union is closed.
This holds for every cell, hence T is closed in S.
Restrict to Sp, and T is closed in Sp.
The topology is right, and Sp is a subcomplex of S.
Direct Limit
The union of all p skeletons gives S.
This is a union where each skeleton builds upon the previous.
If you like category theory, it is technically a
direct limit.
The subset T is closed in S iff it is closed in each skeleton Sp.
This is clear if you remember that T is closed iff its intersection with every cell is closed.
The latter is true iff T is closed in each cell of dimension p or less, for each p -
and this holds iff T is closed in each Sp.