TOP (category)

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Definition

$\mathrm{TOP}$ is the category of all topological spaces, the objects are tuples of a set $X$ and a topology $\mathcal{J}_X$ on $X$ and the arrows, or morphisms of the category are continuous functions. More explicitly.

• The objects of $\mathrm{TOP}$ are all topological spaces, $(X,\mathcal{J}_X)$
• The arrows/morphisms of $\mathrm{TOP}$ are the continuous functions between spaces.

Discussion

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TODO: Discuss as a subcategory of $\mathrm{SET}$, remember it must first go under the forgetful functor to discard the topological structure and distill it to just sets and mappings

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