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  in a neighborhood set  of .

1. There corresponds to each point  at least one neighborhood , and each neighborhood  contains the point .

2. If  and  are two neighborhoods of the same point , there must exist a neighborhood  that is a subset of both.

3. If the point  lies in , there must exist a neighborhood  that is a subset of .

4. For two different points  and , there are two corresponding neighborhoods  and  with no points in common.

REFERENCES:

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