Category of topological spaces

The category of topological spaces, denoted $𝖳𝗈𝗉$ or $𝐓𝐨𝐩$, is the category whose objects are topological spaces and whose morphisms are continuous functions.

Definition

The category $𝖳𝗈𝗉$ consists of:

• $\mathrm{ob}(𝖳𝗈𝗉)$ consists of all topological spaces,
• $\mathrm{mor}(𝖳𝗈𝗉)$ consists of all continuos functions.

Subcategories

Several important subcategories of $𝖳𝗈𝗉$ arise by restricting objects:

• $𝖧𝖺𝗎𝗌$, category of Hausdorff spaces;
• $𝖢𝗈𝗆𝗉$, category of Compact spaces;

• ${𝖳𝗈𝗉}_{*}$, category of pointed topological spaces;
• $𝖢𝖶$, category of CW complexes or cell complexes.