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topology.​category.​Top.​basic

topology.​category.​Top.​basic

source
Top
Top.​bundled_hom
Top.​comp_app
Top.​discrete
Top.​id_app
Top.​inhabited
Top.​of
Top.​topological_space
Top.​topological_space_unbundled
Top.​trivial
source
def Top  :
Type (u+1)

The category of topological spaces and continuous maps.

Equations
source
@[instance]
def Top.​bundled_hom  :
category_theory.bundled_hom continuous_map

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source
@[instance]
def Top.​topological_space_unbundled (x : Top) :
topological_space ↥x

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source
@[simp]
theorem Top.​id_app (X : Top) (x : ↥X) :
⇑(𝟙 X) x = x

source
@[simp]
theorem Top.​comp_app {X Y Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↥X) :
⇑(f ≫ g) x = ⇑g (⇑f x)

source
def Top.​of (X : Type u) [topological_space X] :
Top

Construct a bundled Top from the underlying type and the typeclass.

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source
@[instance]
def Top.​topological_space (X : Top) :
topological_space ↥X

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source
@[instance]
def Top.​inhabited  :
inhabited Top

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source
def Top.​discrete  :
Type u ⥤ Top

The discrete topology on any type.

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source
def Top.​trivial  :
Type u ⥤ Top

The trivial topology on any type.

Equations

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