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

Terminology

en	fr	de	zh		ja
category of topological spaces	catégorie des espaces topologiques	Kategorie der topologischen Räume	拓扑空间范畴	拓撲空間範疇	位相空間の圏
topological space	espace topologique	topologischer Raum	拓扑空间	拓撲空間	位相空間
continuous function	fonction continue	stetige Funktion	连续函数	連續函數	連続関数
category of Hausdorff spaces	catégorie des espaces de Hausdorff	Kategorie der Hausdorff‑Räume	Hausdorff 空间范畴	Hausdorff 空間範疇	ハウスドルフ空間の圏
category of compact spaces	catégorie des espaces compacts	Kategorie der kompakten Räume	紧空间范畴	緊空間範疇	コンパクト空間の圏
category of pointed topological spaces	catégorie des espaces pointés	Kategorie der punktierten topologischen Räume	带点拓扑空间范畴	帶點拓撲空間範疇	点付き位相空間の圏
category of CW complexes	catégorie des complexes CW	Kategorie der CW‑Komplexe	CW复形范畴	CW複形範疇	CW複体の圏