I have following question:
$X, X_0, X_1$ and $X_2$ are topological spaces.
Furthermore, $\mu_i:X_0\rightarrow X_i$ and $\alpha_i:X_i\rightarrow X$ morphisms in the category of topological spaces, $i=1,2,$ such that $\alpha_1\circ\mu_1=\alpha_2\circ\mu_2$. And the following diagram is a pushout:
$$\require{AMScd}
\begin{CD}
X_0 @>\mu_1>> X_1 \\
@V\mu_2VV @VV\alpha_1V \\
X_2 @>\alpha_2>> X
\end{CD}
$$
How can I show that
1) $X=\alpha_1(X_1)\cup \alpha_2(X_2)$,
2) $\alpha_1(X_1)\cap \alpha_2(X_2)=\alpha_1(\mu_1(X_0))=\alpha_2(\mu_2(X_0))$,
3) If $\mu_1$ is injective, so is $\alpha_2$?
Many thanks in advance!
Alex
Best Answer
Hint. The pushout is constructed in such a way that the forgetful functor $\mathsf{Top} \to \mathsf{Set}$ preserves it. So you might as well work with sets and maps of sets. Now use the explicit construction of the pushout as $X=(X_1 \sqcup X_2)/(\mu_1(x) \sim \mu_2(x))$.