What is the difference between a topological and a metric space?
What is the difference between topological and metric spaces?
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$\begingroup$
general-topology
metric-spaces
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8Don't yell! (Capital letters are interpreted as shouts). And please make your titles informative. "I want to learn" is not informative. – 2011-02-10
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0Metric spaces are a specific type of topological space, with many nice properties. They are much easier to understand intuitively, and general enough for many applications. – 2011-02-10
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1Every metric space is a topological space, but not every topological space is a metric space. Do you know the definitions, or are you just now encountering them? – 2011-02-10
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7@Arturo: A nitpick. I have heard this said by many people "Every metric space is a topological space". I would actually prefer to say every metric space induces a topological space on the same underlying set. – 2011-02-10
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5@Sivaram: Fair enough. I say it in much the same sense that I say "Every ring is an abelian group", when I should "really" say "there is a faithful forgetful functor from Rings to AbGroups which commutes with the forgetful functors to Set"... (-: – 2011-02-10