The composition of continuous maps is continuous

Statement

Let $(X,\mathcal{ J })$, $(Y,\mathcal{ K })$ and $(Z,\mathcal{ H })$ be topological spaces (not necessarily distinct) and let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be continuous maps, then:

• their composition, $g\circ f:X\rightarrow Z$, given by $g\circ f:x\mapsto g(f(x))$, is a continuous map.

Consequences and importance of theorem

This theorem is important in that it shows TOP is actually a category, it shows that the composition of morphisms is a morphism.

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TODO: expand on importance

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Proof

Grade: D

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This is really really easy, I could probably write it in the time it has taken me to write this instead. Marked as low-hanging fruit

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