nLab underlying topological space

In topology, the notion of topological space is generally taken as basic, and other related structures (such as a metric, a uniform structure, etc) are typically seen as extra structure on that space. Then it's natural to speak of the underlying topological space of a given metric space etc.

One can take other perspectives. For example, topological spaces, uniform spaces, and metric spaces can all be viewed as quasigauge spaces with extra property, and then there is almost no overlap between these. However, Top is a reflective subcategory of QGau, and the underlying topological space of a metric or uniform space is the reflection in $Top$ of its underlying quasigauge space.

In any case, given any notion of space $X$ with an underlying set ${|X|}$ of points and a frame of open subsets (i.e., a subframe $\mathcal{O}X$ of the power set $\mathcal{P}{|X|}$), we do obtain a topological space $({|X|},\mathcal{O}X)$ that may be said to underlie $X$.