If \(f\colon S^{n-1} \to X\) then
\[
X\cup_f D^n = X\amalg D^n/\sim
\]
is the space obtained by attaching an \(n\)-dimensional cell to \(X\) via the map \(f\).
Example1.7.2
If \(X = \{p\}\) and \(f\colon S^{n-1} \to X\) then
\[
X\cup_f D^n\cong D^n/S^{n-1} \cong S^n.
\]
Definition1.7.3Finite cell complex
A 0-dimensional finite cell complex is a finite disjoint union of points.
A \(k\)-dimensional finite cell complex is a space obtained by attaching finitely many \(k\)-cells to a \((k-1)\)-dimensional finite cell complex.
Example1.7.4
Example1.7.5
If a finite cell complex \(X\) has one 0-cell and one \(n\)-cell then \(X \cong S^n\).
Similarly if \(X\) has one 0-cell and \(k\)\(n\)-cells then \(X \cong \bigvee_{i=1}^{k}S^n\).
Example1.7.6
\(T^2\) is a finite cell complex with one 0-cell, two 1-cells and one 2-cell.
Example1.7.7
Any simplicial complex is a finite cell complex and any closed manifold can be given the structure of a finite cell complex.
