Topological Spaces

Definition: Topology

A topology on a non-empty set $X$ is a collection $\tau$ of subsets of $X$ which has the following properties:

• The empty set $\varnothing$ and $X$ are in $\tau$.

• The union of any subset of $\tau$ is in $\tau$.

• The intersection of any two elements of $\tau$ is in $\tau$.

INTUITION

A topology $\tau$ on a set $X$ can be interpreted as a definition of “closeness” between elements of $X$ without using any notion of distance. Moreover, a topology provides a way for us to define what is “inside” a set, what is “outside” a set and what separates the inside of a set from its outside.

EXAMPLE

Consider the sets $X=\{1,2,3,4,5,6\}$ and $\tau =\{\varnothing ,X,\{1\},\{3,4\},\{1,3,4\},\{2,3,4,5,6\}\}$. The set $\tau$ is a topology on $X$, since it satisfies the requirements in the definition.

Definition: Topological Space

A topological space $(S,\tau )$ is a non-empty set $S$ equipped with a topology $\tau$ on it.

Note: Points

It is common to refer to the elements $s\in S$ of a topological space as points.