TopOlogy

Topology is a branch of mathematics dealing exclusively with properties of continuity. Formally, a topology for a space X is defined a set T of subsets of X satisfying:

1) T is closed under abitrary unions

2) T is closed under finite intersections

3) X, {} are in T

The sets in T are referred to as open sets, and their complements as closed sets. Roughly speaking open sets are thought of as neighborhoods of points. This definition of topology is too general to be of much use and so normally additional conditions are imposed.----

The importance of TopOlogy is, in part, that one can define different topological spaces with elements from AnalySis, AlgeBra, or GeoMetry and then one can determine the properties of such spaces and prove theorems about them. Of equal importance, one can prove what is not true about such spaces. Thus, through TopOlogy one can obtain results in AnalySis, AlgeBra and GeoMetry. This makes TopOlogy very powerful.