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Topological Space

A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions:

1. The empty set emptyset is in T.

2. X is in T.

3. The intersection of a finite number of sets in T is also in T.

4. The union of an arbitrary number of sets in T is also in T.

Alternatively, T may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become:

3. The intersection of an arbitrary number of sets in T is also in T.

4. The union of a finite number of sets in T is also in T.

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 intersection _(n=1)^infty(-1/n,1/n)={0}, where an infinite intersection of open intervals is a closed set.

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).

See also

Closed Set, Hausdorff Axioms, Kuratowski’s Closure-Complement Problem, Manifold, Neighborhood, Open Neighborhood, Open Set, Sigma-Compact Topological Space, T2-Space, Topological Vector Space, Topology Explore this topic in the MathWorld classroom

Portions of this entry contributed by Johannes Lipp

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References

Berge, C. Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. New York: Dover, 1997.Hausdorff, F. Grundzüge der Mengenlehre. Leipzig, Germany: von Veit, 1914. Republished as Set Theory, 2nd ed. New York: Chelsea, 1962.Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

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Topological Space

Cite this as:

Lipp, Johannes and Weisstein, Eric W. “Topological Space.” From MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/TopologicalSpace.html

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