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A topological space, also called an abstract topological space, is a set
1. The empty set
2.
3. The intersection of a finite number of sets in
4. The union of an arbitrary number of sets in
Alternatively,
3. The intersection of an arbitrary number of sets in
4. The union of a finite number of sets in
These axioms are designed so that the traditional definitions of open and closed intervals of the real line continue to be true. For example, the restriction in (3) can be seen to be necessary by considering
In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff axioms (which in modern times are not considered necessary in the definition of a topological space). (责任编辑：) |
