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:

• $\mathrm{ob}(𝖳𝗈𝗉)$ is the class of all topological spaces;
• $\mathrm{mor}(𝖳𝗈𝗉)$ is the class of all continuous functions between topological spaces;
• The composition operation $\circ$ is given by the composition of ordinary 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.

See also

• Category theory
• Morphism
• Category of sets
• Category of groups