Hoping to get a better understanding of CW complexes in terms of pushouts, I have decided to take a look at Jeffrey Strom's Modern Classical Homotopy Theory (truly wonderful title for a book, by the way), and I pretty much immediately manage to find myself dumbfounded. The question in particular is Problem 3.1:
Let $i: S^{n-1} \rightarrow D^n$ be the inclusion of the boundary. (a) Show that the functions $j_N : x \mapsto (x, \sqrt{1-|x|^2})$ and $j_S : x \mapsto (x, - \sqrt{1-|x|^2})$ are homeomorphisms. (b) Show that the diagram (*) is a pushout square in the category Top.
The diagram being the following one.
Part (a) is no difficulty, but on part (b) I'm so hopelessly lost that I cannot even figure out where to begin!
Any and all help is appreciated.
The pushout of CW complexes (or most things in general) $X \xleftarrow{i} Z \xrightarrow{j} Y$ is a quotient of the disjoint union $X \coprod Y / \sim$ where $x \sim y$ if there is a $z \in Z$ such that $i(z) = j(z)$. CW complexes are closed under taking quotients so everything is fine here.
$j_S$ and $j_N$ map disks $D^{n+1}$ to the Southern (resp. Northern) "hemispheres" of $S^{n+1}$. Hence, the pushout is the disjoint union of two $D^{n+1}$ such that their equators are identified. This, of course, is just $S^{n+1}$.
To do this properly, you would want to show that $j_S \circ i = j_N \circ i$ and for any CW $W$ and maps $D^{n+1} \xrightarrow{j_N'} W$, $D^{n+1} \xrightarrow{j_S'} W$, that there is a unique $g: S^{n+1} \rightarrow W$. In some sense this is saying that the pushout $S^{n+1}$ is the largest object such that the diagram commutes.
By the way, $j_S$ and $j_N$ are not homeomorphisms. But, in any case they are homeomorphisms onto their images.