Franklin's Notes

Topological space

A topological space is a set $X$ together with a collection $\mathcal{T}\subset 2^X$ of distinguished subsets of $X$ called open sets, required to satisfy the following properties:

1. The empty set $\varnothing$ and the set $X$ are open subsets of $X$, so that $\varnothing,X\in\mathcal{T}$
2. The union of arbitrarily (possibly infinitely) many open sets is always open, so that $\mathcal T$ is closed under arbitrary unions
3. The intersection of finitely many open sets is always open, so that $\mathcal T$ is closed under finite intersections

Intuitively, the definition of an open set is meant to formalize our metaphysical notion of "nearness", but without relying on a concept of distance, as metric spaces do. (Every metric space gives rise to a topology , but not every topology is metrizable.) An open set can be thought of informally as "a set of points that can be grouped together by closeness". Also, the subsets of $X$ that are complements of an open set are called closed sets.

Given two topologies $\mathcal T$ and $\mathcal T'$ on the same set $X$ such that $\mathcal T\subset\mathcal T'$, we say that $\mathcal T$ is coarser than $\mathcal T'$, or equivalently, $\mathcal T'$ is finer than $\mathcal T$. Intuitively, in a finer topology, it is possible to make "more discriminations" between sets of points. The finest topology on a set $X$ is the discrete topology which is generated by a basis of singleton sets, and the coarsest topology on $X$ is the indiscrete topology, which is generated by ${X}$. (These two constructions, it happens, are related via an interesting adjunction chain .)

Every topological space gives rise to a lattice , namely the lattice of open sets. The meet of two open sets is their intersection, and the join is their union. (However, not every lattice is induced by a topology in this way.)


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