A closed embedding, mapping cylinder

Question

In tom Dieck's algebraic topology, pg81, he defines:

He claims that as $\langle i_0,i_1\rangle$ is a closed embedding, the maps $\langle j, J \rangle, j, J$ are closed embeddings too. How does this follow ?

Answer 1

You can see this directly as the pushout is nothing else than the quotient space of the disjoint sum $X \times I + Y$ with respect to $f(x) \sim (x,1)$.

You can also use Proposition (1.2.4) in tom Dieck's book to see that $\langle j, J \rangle$ is a closed embedding. But obviously the inclusions $i_X : X \to X + Y$ and $i_Y : Y \to X + Y$ are closed embeddings, hence also $j = \langle j, J \rangle \circ i_X$ and $J = \langle j, J \rangle \circ i_Y$ are.

Now you might demur that (1.2.4) is stated without proof. However, the proof is really straightforward. See for example Proposition (1.8.1) in

tom Dieck, General topology (reference [45]).