Topological space

The definition separates two ingredients: the set of points \(X\) and the family \(\mathcal{T}\) that says which subsets count as open. The same set can carry many different topologies, and changing \(\mathcal{T}\) changes the space, not the set. That is why the pair \((X,\mathcal{T})\) is written, and one says «the space \(X\)» only when the topology is understood.

The everyday examples are \(\mathbb{R}\) with the usual topology, whose open sets are unions of intervals \((a,b)\), and any metric space with the open sets defined by its distance. But there are topological spaces that come from no metric, such as a set whose only open sets are \(\varnothing\) and \(X\), which is why the notion is more general.

A common trap is to believe that a set «without boundary» is open in any space. Being open depends on the chosen topology: \(\{1\}\) is not open in the usual \(\mathbb{R}\), but it is in \(\mathbb{Z}\) with the discrete topology. The axioms guarantee the minimum: \(\varnothing\) and \(X\) are always open, and arbitrary unions and finite intersections of open sets stay open.

From the open sets one defines closed sets, neighborhoods, interior and closure, and continuous maps. The next concepts of this topic first go through the two extreme cases, the discrete and the trivial topology, and then those constructions.