A topological space, also called an abstract topological space, is a set 
together with a collection 
of subsets, called open
sets, satisfying the following 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 intervals 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 
, 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).
