Topology: An Inquiry-Based Approach

Many of the properties that we introduced in metric spaces (continuity, limit points, boundary) could be phrased in terms of the open sets in the space. With that in mind, we can broaden our concept of space by eliminating the metric and just defining the opens sets in the space. This produces what are called topological spaces.

Recall that the open sets in a metric space satisfied certain properties, including that the arbitrary union of open sets is open and any finite intersection of opens sets is open. We will now take these properties as our axioms in defining topological spaces.

A topological space is then any set on which a topology is defined. If $X$ is the space and $\tau$ a topology on $X\text{,}$ we denote the topological space as $(X,\tau )\text{.}$ The elements of $\tau$ are called the open sets in the topological space. When the topology is clear from the context, we simple refer to $X$ as the topological space. Some examples are in order.