TOP (category)
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Definition
[ilmath]\mathrm{TOP} [/ilmath] is the category of all topological spaces, the objects are tuples of a set [ilmath]X[/ilmath] and a topology [ilmath]\mathcal{J}_X[/ilmath] on [ilmath]X[/ilmath] and the arrows, or morphisms of the category are continuous functions^{[1]}. More explicitly.
 The objects of [ilmath]\mathrm{TOP} [/ilmath] are all topological spaces, [ilmath](X,\mathcal{J}_X)[/ilmath]
 The arrows/morphisms of [ilmath]\mathrm{TOP} [/ilmath] are the continuous functions between spaces.
Discussion
TODO: Discuss as a subcategory of [ilmath]\mathrm{SET} [/ilmath], remember it must first go under the forgetful functor to discard the topological structure and distill it to just sets and mappings
References

