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Topological Space
A topological space is a space where the concept of openness is defined.
Formal definition
A ordered pair $(X, \tau)$ is topological space where $X$ is a set and $\tau$ is a collection of subsets of $X$ which are considered open and has to satisfy the following
- $\phi, X \in \tau$
- Closed under arbitrary (possibly infinite) union: $\displaystyle \bigcup_{t \in \tau}^\infty t \in \tau$
- Closed under finite intersection: $\displaystyle \bigcap_{t \in \tau}^{i} t \in \tau$
Basis
You might notice that defining this $\tau$ could be a pain-in-the-ass. You might also notice that a smaller subset of $\tau$ could possibly enumerate all of $\tau$. This is the basis for basis (I’m funny).