I was recently introduced to category theory and am looking to verify (proof?) a universal property from a diagram as shown.
In this example, the description says: Take a topological space covered by two open subsets: $X = U \cup V$. The diagram of inclusion maps has a universal property in the world of topological spaces and continuous maps.
Attempt at verifying a universal property:
Let $f: U → Y$ and $g : V → Y$ be any continuous functions so that $f \circ i$ = $g \circ j$. Then there exists a unique continuous function $h : X → Y$ such that $h \circ i' =g$ and $h \circ j' = f$ so that the diagram commutes.
I'd appreciate if someone could guide me on the right track towards my first verification of universal property.
If you have continuous functions $f: U\to Y$ and $g: V\to Y$ you can define $h:U\cup V\to Y$ as follows $$ h(x)=\begin{cases} f(x) & x\in U\\ g(x) & x\in V \end{cases} $$ I'll leave it to you to show that, given the conditions of your problem, $h$ is well-defined and continuous.
In the more general language of category theory, your diagram is a special case of a pushout (look here) in the category of topological spaces.