The composition of continuous maps is continuous

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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


Grade: D
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  1. Introduction to Topological Manifolds - John M. Lee