**4**

Topology

**4.1 Topology, topological spaces**

Abstracting away from metrics, a topological space is a set, with a collection of so-called *open subsets U*, satisfying the following properties. We have already seen them, for sequentially open subsets of a metric space, in Proposition 3.2.7.

**Definition 4.1.1** (Topology) Let *X* be a set. A *topology* on *X* is a collection of subsets of *X*, called the *opens* of the topology, such that:

- every union of opens is open (including the empty union, Ø);
- every finite intersection of opens is open (including the empty intersection, taken as
*x*itself).

A *topological space* is a pair (*X*, ), where is a topology on *X*.

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