Topology and Topological Space
WHAT A SET NEEDS BEFORE CONTINUITY MEANS ANYTHING
A set is a bag of points and nothing more, with no notion of which maps out of it are continuous. A topology is the least structure that supplies one: a choice of which subsets to call open, made once, and everything else in the subject is built on top of that choice.
The definition
Let \(M\) be a set. A topology \(\mathcal{O}\) on \(M\) is a subset of the powerset, \(\mathcal{O} \subseteq \mathcal{P}(M)\), satisfying:
- \(\emptyset \in \mathcal{O}\) and \(M \in \mathcal{O}\);
- given \(U, V \in \mathcal{O}\), then \(U \cap V \in \mathcal{O}\);
- let \(A\) be an arbitrary index set; given \(U_\alpha \in \mathcal{O}\), then \(\bigcup_{\alpha \in A} U_\alpha \in \mathcal{O}\).
The members of \(\mathcal{O}\) are called the open sets, and a set is open for no reason other than belonging to \(\mathcal{O}\). Only \(\emptyset\) and \(M\) are open in every topology there is, put there by condition (i).
The asymmetry in the last two conditions
Condition (ii) is about two sets, and induction extends it to any finite number. Condition (iii) allows any family at all, however large. Finite for intersections, arbitrary for unions, and the difference is not fussiness: on \(\mathbb{R}\) with its standard topology the intervals \(\left(-\tfrac{1}{n}, \tfrac{1}{n}\right)\) are all open, and \(\bigcap_{n=1}^{\infty} \left(-\tfrac{1}{n}, \tfrac{1}{n}\right) = \{0\}\), which is not.
On \(\{1,2,3\}\), the collection \(\big\{\emptyset, \{1\}, \{2\}, \{1,2,3\}\big\}\) fails (iii): \(\{1\} \cup \{2\} = \{1,2\}\) is not in it.
The pair is the object
Let \(M\) be a set and \(\mathcal{O}\) a topology on \(M\). The double \((M, \mathcal{O})\) is called a topological space.
A set of more than one point carries many topologies, and they are different structures rather than different descriptions of one. Which is why the pair travels together: \(M\) alone does not determine what is open, so it does not determine which maps are continuous.