PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Topological spaces and continuous maps form the large locally small category $\mathbf{Top}$

Statement

Topological spaces and continuous maps form a large locally small category $\mathbf{Top}$ .

Facts & Assumptions

Given: Topological spaces $X,Y,Z$ and continuous maps $f:X\to Y$ , $g:Y\to Z$ .

[L1]

A topology is the structure in Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, and identity maps and composites of continuous maps are continuous by the inverse-image definition in Continuity of a map of topological spaces at a point and globally.

[L2]

The functions $A\to B$ between fixed sets form the set $B^A$ (The set $B^{A}$ of all functions $A \to B$ ); category size is as in Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, and the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).

Proof

technique · direct

1.1

Identity maps and $g\circ f$ are continuous by [L1], while associativity and unit equations are those of function composition.

givenL1

2.1

Consequently spaces and continuous maps form a category, and each hom-collection is a set of functions between fixed underlying sets.

step 1.1L2

3.1

Every singleton $\{\alpha\}$ , for an ordinal $\alpha$ , has its unique topology; these give distinct space objects, so [L2] makes $\mathbf{Top}$ large and locally small.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)

Topological spaces and continuous maps form the large locally small category Top\mathbf{Top}Top