Let $(X,\delta)$ be a metric space,$(Y,\tau)$ a topological space.Assume that $(X,\delta)$ is homeomorphic to $(Y,\tau)$ .
Is $(Y,\tau)$ metrisable ,i.e there is a metric that induces the topology on $Y$.
If false ,can you give me an example?
Topological space +metrization + homeomorphism
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$\begingroup$
general-topology
metric-spaces
2 Answers
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If $(X,d)$ is a metric space, $(Y,\tau)$ is a topological space and $\phi : X \to Y$ is a homeomorphism, then $d^*(y_1,y_2) := d(\phi^{-1}(y_1), \phi^{-1}(y_2))$, for $y_1,y_2 \in Y$ defines a metric on $Y$ that generates the correct topology. That is, $(Y,d^*)$ is a metric space with topology $\tau$.
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A homeomorphism tells you that two topological spaces are the "same". So, yes intuitively $(Y,\tau)$ should have a metric space structure as well.
To get this structure, let $\varphi:Y\to X$ be our homeomorphism. Define a metric $d$ on $Y$ by $d(y_1,y_2)=\delta(\varphi(y_1),\varphi(y_2))$. Try to show that this is indeed a metric on $Y$.