# Metrizable space

Somebody please explain me what is meant by a space X is metrizable?

What is the relationship between metrizable and compactness, connectednes, regularity, normality and Hausdorffness?

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Had you read the article of wikipedia on metrizability? – Seirios Dec 24 '12 at 9:10

Metrizable is a topological space homeomorphic to a metric space. That is $(X,\mathcal{T})$ is metrizable when there exists a metric $d:X^2\to [0,+\infty)$ so that the topology $\mathcal{T}_d$ induced by $d$ coincides with $\mathcal{T}$. The spaces $(X,\mathcal{T})$ and $(X,\mathcal{T}_d)$ are homeomorphic. As any metric space is Hausdorff, every metrizable space is also Hausdorff.