The composition of continuous maps is continuous
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Statement
Let [ilmath](X,\mathcal{ J })[/ilmath], [ilmath](Y,\mathcal{ K })[/ilmath] and [ilmath](Z,\mathcal{ H })[/ilmath] be topological spaces (not necessarily distinct) and let [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]g:Y\rightarrow Z[/ilmath] be continuous maps, then^{[1]}:
 their composition, [ilmath]g\circ f:X\rightarrow Z[/ilmath], given by [ilmath]g\circ f:x\mapsto g(f(x))[/ilmath], 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.
TODO: expand on importance
Proof
Grade: D
This page requires one or more proofs to be filled in, it is on a todo list for being expanded with them.
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The message provided is:
This proof has been marked as an page requiring an easy proof
The message provided is:
This is really really easy, I could probably write it in the time it has taken me to write this instead. Marked as lowhanging fruit
This proof has been marked as an page requiring an easy proof
References
