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What’s The Deal With Hausdorff Spaces? (Part I) | Notational Notions

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If you have spent more than a few minutes with point-set topology, chances are that you have heard the term “Hausdorff space.” The axioms of a topological space are perhaps a little too general, and so most topological theorems impose additional axioms. Here are some of the basic separation axioms: A space is called if for any two distinct points and , there exists an open set that contains exactly one of and . A space is called if for any two distinct points and , there exists an open set that contains but not , and an open set that contains but not . A space is called , or Hausdorff, if for any two distinct points and , there exist disjoint open sets and such that contains and contains . It should be clear that Hausdorff spaces are , and spaces are . One example of a space that is not Hausdorff is the integers, where is an open set if is finite. There are many, many other separation axioms, in fact, there are increasingly strong axioms called , , , and . So why is

What’s The Deal With Hausdorff Spaces? (Part I) | Notational Notions Notational Notions confessions of a mathoholic Home About What’s The Deal With Hausdorff Spaces? (Part I) If you have spent more than a few minutes with point-set topology, chances are that you have heard the term “Hausdorff space.” The axioms of a topological space are perhaps a little too general, and so most topological theorems impose additional axioms. Here are some of the basic separation axioms: A space is called if for any two distinct points and , there exists an open set that contains exactly

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