Basis for a Topology

Generate a topology from a basis and compare two topologies.

The idea

A basis is a compact way to specify a topology: instead of listing every open set, we name a small stock of standard ones and declare the rest to be their unions. The usual topology on $\mathbb{R}$ shows the need — it has uncountably many open sets, so no list of them can be written down, but every one is a union of open intervals, and naming the intervals specifies the whole topology.

Let $\mathcal{B}$ be a collection of subsets of $X$, and declare a set $U$ open when for each point $x \in U$ some member $B \in \mathcal{B}$ satisfies $x \in B \subseteq U$ — equivalently, when $U$ is a union of members of $\mathcal{B}$. Two axioms make this a topology. First, every point of $X$ must lie in some member, so that $X$ itself comes out open. Second, whenever $x$ lies in an overlap $B_1 \cap B_2$, some member $B_3$ must satisfy $x \in B_3 \subseteq B_1 \cap B_2$, so that the intersection of two open sets comes out open, as the topology axioms require. A collection satisfying both is a basis, its members are basis elements, and the topology described is the topology generated by $\mathcal{B}$.

A basis is not unique: different collections can generate the same topology. This makes bases the practical way to compare topologies — we compare two small bases rather than the topologies themselves.

Ways to work on it

• Walkthrough. The basis axioms and the topology a basis generates.
• Practice. Count the open sets generated by a small basis.
• Hardest. Use the comparison lemma to decide which topology is finer.

Not sure where to start? Take the ten-question placement test.