Topological Space
A topological space, also called an abstract topological space, is a set 
together with a collection of open subsets 
that satisfies the four conditions:
1. The empty set 
is in 
.
2. 
is in 
.
3. The intersection of a finite number of sets in 
is also in 
.
4. The union of an arbitrary number of sets in 
is also in 
.
Alternatively, 
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 
is also in 
.
4. The union of a finite number of sets in 
is also 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 ![intersection _(n=1)^infty(-1/n,1/n)=<span style=]()
{0}" src="https://mathworld.wolfram.com/images/equations/TopologicalSpace/Inline16.gif" style="height:17px; width:135px" />, 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).
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.
