Topology from subspace metric equals subspace topology: Difference between revisions

Latest revision as of 16:21, 26 October 2009

Statement

Statement with symbols

Suppose $(X, d)$ is a metric space. Then, we can consider the induced topology on $X$ from the metric.

Now, consider a subset $Y$ of $X$ . The metric on $X$ induces a Subspace metric (?) on $Y$ , by restriction. Thus, there are two possible topologies we can put on $Y$ :

The Subspace topology (?) from the topology induced by the metric on $X$
The induced topology from the subspace metric on $Y$

These two topologies are the same.

Definitions used

Topology induced by a metric

Subspace topology

Proof

Fill this in later

@@ Line 5: / Line 5: @@
 Suppose <math>(X,d)</math> is a [[metric space]]. Then, we can consider the [[metric induces topology|induced topology]] on <math>X</math> from the metric.
-Now, consider a subset <math>Y</math> of <math>X</math>. The metric on <math>X</math> induces a metric on <math>Y</math>, by restriction. Thus, there are two possible topologies we can put on <math>Y</math>:
+Now, consider a subset <math>Y</math> of <math>X</math>. The metric on <math>X</math> induces a [[fact about::subspace metric]] on <math>Y</math>, by restriction. Thus, there are two possible topologies we can put on <math>Y</math>:
-* The [[subspace topology]] from the topology induced by the metric on <math>X</math>
+* The [[fact about::subspace topology]] from the topology induced by the metric on <math>X</math>
 * The induced topology from the subspace metric on <math>Y</math>