Franklin's Notes

Basis for a topology

In a topological space $(X,\mathcal T)$, a basis is a subset $\mathcal B\subset \mathcal T$ of open sets that generate all other open sets. That is, $\mathcal B\subset \mathcal T$ is a basis if and only if every element of $\mathcal T$ is a union of some collection of the basis elements in $\mathcal B$. The elements of $\mathcal B$ are sometimes called basic open sets.

The following three conditions are together equivalent to a set of subsets $\mathcal B\subset 2^X$ comprising a basis for some topology on $X$:

1. Every element of $X$ is contained in some basic open set of $\mathcal B$.
2. For every $B_1,B_2\in\mathcal B$ and every $x\in B_1\cap B_2$, there exists $B_3\in\mathcal B$ such that $x\in B_3\subset B_1\cap B_2$.

Here's a way of visualizing the second condition:

The necessity of the first condition is clear, since $X$ is always an open set and therefore $X$ must be a union of some set of basic open sets. The necessity of the second condition follows from the fact that if $B_1,B_2$ are open, then their intersection $B_1\cap B_2$ must be open, meaning that it is a union of basic open sets, and therefore each point therein is contained in at least one basic open set. To see why these conditions are sufficient, notice that by construction, arbitrary unions and finite intersections of any union of elements of $\mathcal B$ are also unions of elements of $\mathcal B$, assuming that the above $2$ properties are satisfied.

A basis is called an irreducible basis, or just irreducible, if no basis element of $\mathcal B$ can be expressed as a union of other basis elements.


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