Category of pointed topological spaces

The category of pointed topological spaces or base spaces $𝖳 𝗈 𝗉 • ≃ ∙ ↓ 𝖳 𝗈 𝗉$ consists of topological spaces with a basepoint preserving continuous maps as its morphisms. #m/def/topology This is a full subcategory of the category of topological spaces with base sets $𝖳 𝗈 𝗉 Ω$ consisting of topological spaces with a subset of base points as objects and base set preserving continuous maps as its morphisms, #m/def/topology i.e. if $𝑓 \in 𝖳 𝗈 𝗉 Ω (𝑋 𝐴, 𝑌 𝐵)$ then $𝑓 (𝐴) \subseteq 𝐵$ .

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