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How is the usual topology on the real number line built from open intervals?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, the usual topology on the real number line is constructed by defining the collection of open sets as all possible unions of open intervals.1

Alternatively, this topology can be characterized by the condition that a set $U$ of real numbers is open if and only if, for every point $p$ in $U$ , there exists an open interval centered at $p$ with a positive radius that is entirely contained within $U$ .2

This structure satisfies the fundamental axioms of a topological space:
* The real line itself and the empty set are open.12
* The intersection of any finite number of open sets is open.12
* The union of any collection of open sets is open.12

Because any two distinct points on the real line can be contained within disjoint open intervals, this construction also ensures that the real line functions as a Hausdorff space.2