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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

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up vote 6 down vote accepted

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.

Proving that a topological space is metrizable lets us in effect consider that space a metric space. Then we can define completeness of the space etc. For more information go here (as Seirios suggested). Now the relationship between metrizable and compactness etc. is the same as the relationship between metric spaces and compactness etc.

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Typo error @ homeorphic. –  Haskell Curry Dec 24 '12 at 9:16
    
@HaskellCurry Fixed. Thank you ( I always make typos when typing words like that) –  Nameless Dec 24 '12 at 9:17

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