Compact to Hausdorff implies closed: Difference between revisions

Latest revision as of 06:12, 23 December 2009

Statement

Any continuous map from a compact space to a Hausdorff space is a closed map i.e. the image of any closed set is closed.

Applications

Any surjective continuous map from a compact space to a Hausdorff space is a quotient map
Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding

Facts used

Compactness is weakly hereditary: Any closed subset of a compact space is compact in the subspace topology.
Compactness is continuous image-closed: The image of a compact space under a continuous map is a compact space.
Hausdorff implies KC: Any compact subset of a Hausdorff space is closed.

Proof

Given: A compact space $X$ , a Hausdorff space $Y$ , a continuous map $f : X \to Y$ .

To prove: For any closed subset $A$ of $X$ , $f (A)$ is a closed subset of $Y$ .

Proof:

$A$ is compact under the subspace topology: This follows from the given datum that $X$ is compact and fact (1).
$f (A)$ is compact under the subspace topology in $Y$ : First, note that the map $f |_{A} : A \to Y$ is continuous, because it is the composite of the inclusion of $A$ in $X$ with the map $f$ , both of which are continuous. Thus, by fact (2), $f (A)$ is compact with the subspace topology from $Y$ .
$f (A)$ is closed in $Y$ : This follows from the previous step, the given datum that $Y$ is Hausdorff, and fact (3).

@@ Line 1: / Line 1: @@
 ==Statement==
-Any [[continuous map]] from a [[compact space]] to a [Hausdorff space]] is a [[closed map]] i.e. the image of any closed set is closed.
+Any [[continuous map]] from a [[compact space]] to a [[Hausdorff space]] is a [[closed map]] i.e. the image of any closed set is closed.
 ==Applications==
@@ Line 7: / Line 7: @@
 * [[Surjection from compact to Hausdorff implies quotient|Any surjective continuous map from a compact space to a Hausdorff space is a quotient map]]
 * [[Injection from compact to Hausdorff implies embedding|Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding]]
+==Facts used==
+# [[uses::Compactness is weakly hereditary]]: Any closed subset of a compact space is compact in the subspace topology.
+# [[uses::Compactness is continuous image-closed]]: The image of a compact space under a continuous map is a compact space.
+# [[uses::Hausdorff implies KC]]: Any compact subset of a Hausdorff space is closed.
 ==Proof==
-We need to show that given any closed subset of the compact space, its image is closed in the Hausdorff space.
+'''Given''': A compact space <math>X</math>, a Hausdorff space <math>Y</math>, a continuous map <math>f:X \to Y</math>.
+'''To prove''': For any closed subset <math>A</math> of <math>X</math>, <math>f(A)</math> is a closed subset of <math>Y</math>.
-We reason in the following steps:
+'''Proof''':
-* The closed subset of the compact space, is itself compact in the [[subspace topology]]. This is because any closed subset of a compact space is compact. {{proofat|[[Compactness is weakly hereditary]]}}
+# <math>A</math> is compact under the subspace topology: This follows from the given datum that <math>X</math> is compact and fact (1).
-* Thus, the image of this subset, in  the target space, is also compact in the [[subspace topology]]. This follows from the fact that an image of a compact space under a continuous map is also compact. {{further|[[Compactness is continuous image-closed]]}}
+# <math>f(A)</math> is compact under the subspace topology in <math>Y</math>: First, note that the map <math>f|_A: A \to Y</math> is continuous, because it is the composite of the inclusion of <math>A</math> in <math>X</math> with the map <math>f</math>, both of which are continuous. Thus, by fact (2), <math>f(A)</math> is compact with the subspace topology from <math>Y</math>.
-* The image of this subset is a compact subset of a Hausdorff space, hence is closed in the Hausdorff space. This follows from the fact that any compact subset of a Hausdorff space is closed. {{further|[[Hausdorff implies KC]]}}
+# <math>f(A)</math> is closed in <math>Y</math>: This follows from the previous step, the given datum that <math>Y</math> is Hausdorff, and fact (3).