Take a continuous map from a product topology, and fix one of the 2 arguments. Is the resultant map continuous?

Question

Let $A,B$ and $C$ be topological spaces. Suppose we have a continuous map from the product space $$f: A\times B \to C$$ Let $a\in A$, and consider the function $$g_a : B \to C \\ g_a(b) := f(a,b)$$ (Aside - is there a standard name for $g_a$, like a 'section' or something?

Given $f$ is continuous, is $g_a$ also continuous? Is this true for general topologies on $A$, or are some separation conditions required?

Answer 1

Sure, because if you define $i:\{a\}×B\hookrightarrow A×B$, then $i$ is continuous*. Now let $h:B\to\{a\}×B$ be the obvious homeomorphism. Then $g_a=f\circ i\circ h$ is then a composition of continuous maps.

*$i$ is called the inclusion of $\{a\}×B$ into $A×B$. And it's continuous if you put the subspace topology on $\{a\}×B$.

Answer 2

Yes, this is continuous: $g_a=f\circ i_a$ where $i_a:B\to A\times B$ takes $b$ to $(a,b)$. The map $i_a$ is continuous (the inverse image of a standard basis element of $A\times B$ is open in $B$) and therefore so is $g_a$, since composites of continuous maps are continuous.