category of Hausdorff spaces
- notation:
- objects: Hausdorff spaces
- morphisms: continuous functions
- Related categories: ,
- nLab Link
This is the full subcategory of consisting of those spaces that are Hausdorff.
Satisfied Properties
Properties from the database
- is co-Malcev
- is cocomplete
- has equalizers
- has a generator
- is infinitary extensive
- is locally small
- has products
- is strongly connected
- is well-copowered
- is well-powered
Deduced properties
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- is complete
- has cofiltered limits
- has connected limits
- is finitely complete
- has wide pullbacks
- has pullbacks
- has sequential limits
- is Cauchy complete
- has coproducts
- is extensive
- has finite coproducts
- has a strict initial object
- has an initial object
- has disjoint finite coproducts
- has disjoint coproducts
- is distributive
- is infinitary distributive
- is lextensive
- is locally essentially small
- has a generating set
- is inhabited
- has connected colimits
- has filtered colimits
- has directed colimits
- is finitely cocomplete
- has wide pushouts
- has countable coproducts
- has binary coproducts
- has sequential colimits
- has coequalizers
- has pushouts
- has directed limits
Unsatisfied Properties
Properties from the database
- is not balanced
- is not cartesian closed
- is not Malcev
- does not have a regular subobject classifier
- is not skeletal
- does not have a strict terminal object
Deduced properties*
- is not mono-regular
- is not a groupoid
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not discrete
- is not essentially discrete
- is not trivial
- is not thin
- is not pointed
- does not have zero morphisms
- does not have biproducts
- is not left cancellative
- is not preadditive
- is not additive
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not an elementary topos
- does not have a subobject classifier
- is not right cancellative
- is not locally cartesian closed
- is not a Grothendieck topos
- is not unital
- does not have disjoint finite products
- does not have disjoint products
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
- is not epi-regular
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
There are 10 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!
Special objects
- terminal object: singleton space
- initial object: empty space
- products: direct product with the product topology
- coproducts: disjoint union with the disjoint union topology
Special morphisms
- isomorphisms: homeomorphisms
- monomorphisms: injective continuous maps
- epimorphisms: continuous maps with dense image
- regular monomorphisms: embeddings with closed image
- regular epimorphisms: