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Hausdorff Axioms


The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements x in a neighborhood set E of x.

1. There corresponds to each point x at least one neighborhood U(x), and each neighborhood U(x) contains the point x.

2. If U(x) and V(x) are two neighborhoods of the same point x, there must exist a neighborhood W(x) that is a subset of both.

3. If the point y lies in U(x), there must exist a neighborhood U(y) that is a subset of U(x).

4. For two different points x and y, there are two corresponding neighborhoods U(x) and U(y) with no points in common.


See also

T2-Space, Topological Space

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References

Hausdorff, F. Grundzüge der Mengenlehre. Leipzig, Germany: von Veit, 1914. Republished as Set Theory, 2nd ed. New York: Chelsea, 1962.

Referenced on Wolfram|Alpha

Hausdorff Axioms

Cite this as:

Weisstein, Eric W. "Hausdorff Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HausdorffAxioms.html

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