Question 3

Question

What is a topological space? Examples?

Answer

A Topological Space $X$ is a set with a Topology $\tau$.

A topology has $X$ and $\varnothing$.
Any union ($\cup$) of $\tau$ will be in $\tau$
Any finite intersection ($\cap$) of sets from $\tau$ will be in $\tau$.

Example

$X$ is Hausdorff if for $x\ne y\exists \text{neighbourhoods}A,B$ of $x$ and $y$ respectively, such that $A\cap B=\varnothing$.

Basically, sets do not intersect on the topological space, making everything an open set.