1. Topological spaces

These facts suggest that open sets could be used as a starting concept. The resulting theory is more general than the theory of metric spaces and is helpful for various spaces of functions that are not metric.

Definition. A topology on a set $X$ is a family ${\cal T}$ of subsets of $X$ such that ${\cal T}$ is closed under finite intersections and arbitrary unions. The members of ${\cal T}$ are called open sets (or simply open sets) and $X$ with $T$ is a topological space.

We would also list that the the empty set and $X$ itself are open, but these facts already follow from the definition since the union of no sets is empty and the intersection of no subsets is $X$.

All the concepts from Section can be turned into definitions. Many theorems that you know from metric spaces remain valid. However, the definition of a topological space is mismatched with metric spaces in two ways:

• The definition of a topological space is weak, in that it does not ensure that open sets can distinguish between points. In fact, given any set $X$ we can define a rudimentary topology in which the only open sets are the empty set and $X$; in this topology, it is no longer true that a convergent sequence has a unique limit, for example. Therefore we often go beyond the definition of a topological space and impose separation conditions, as in § below.

• Sequences (indexed by nonnegative integers), which are sufficient to determine the topology in a metric space, are inadequate to do so for topological spaces in general. In other words, it is not true in general that a subset $S$ is closed when any convergent sequence in $S$ converges to a point of $S$. Instead, one can use ``nets''--generalized sequences indexed by a directed set instead. However, theorems in topology can usually be proved without using nets.