theorem
closedSubset_compactSpace
{X : Type u_1}
[TopologicalSpace X]
[CompactSpace X]
{Z : Set X}
(hZ : IsClosed Z)
:
CompactSpace ↑Z
Lec 7 / Lem 19: a closed subset of a compact space is compact.
theorem
isClosed_image_of_complete
{X : Type u_1}
{Y : Type u_2}
[TopologicalSpace X]
[TopologicalSpace Y]
[CompactSpace X]
[T2Space Y]
{f : X → Y}
(hf : Continuous f)
{Z : Set X}
(hZ : IsClosed Z)
:
Lec 7: the image of a closed set under a continuous map from a compact space to a Hausdorff space is closed (topological analogue of "proper image closed").
theorem
isClosed_range_of_complete
{X : Type u_1}
{Y : Type u_2}
[TopologicalSpace X]
[TopologicalSpace Y]
[CompactSpace X]
[T2Space Y]
{f : X → Y}
(hf : Continuous f)
:
The range of a continuous map from a compact space to a Hausdorff space is closed.
theorem
prod_compactSpace
{X : Type u_1}
{Y : Type u_2}
[TopologicalSpace X]
[TopologicalSpace Y]
[CompactSpace X]
[CompactSpace Y]
:
CompactSpace (X × Y)
The product of two compact spaces is compact.