0. Facts to recall from metric spaces

First, recall that in a metric space, the family of open sets is closed under finite intersections and arbitrary unions.

Second, recall that in a metric space, a number of concepts can be characterized in terms of open sets (where as usual, we often refer to subsets as simply ``sets''):

• A set is closed $\Leftrightarrow$ its complement is open.

• A function is continuous $\Leftrightarrow$ the inverse image of every open set is open.

• A space is connected $\Leftrightarrow$ it cannot be partitioned into two open sets.

• A space is compact $\Leftrightarrow$ every open cover (family of open sets whose union is the space) has a finite subcover (finite subfamily whose union is the space).

• A point is isolated $\Leftrightarrow$ it is open as a singleton.

• A sequence converges to $x$ $\Leftrightarrow$ for each open set $O$ containing $x$, the sequence is eventually in $O$.