Editing Subspace topology (section)

== Terminology ==

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever <math>S</math> is a subset of <math>X</math>, and <math>(X, \tau)</math> is a topological space, then the unadorned symbols "<math>S</math>" and "<math>X</math>" can often be used to refer both to <math>S</math> and <math>X</math> considered as two subsets of <math>X</math>, and also to <math>(S,\tau_S)</math> and <math>(X,\tau)</math> as the topological spaces, related as discussed above. So phrases such as "<math>S</math> an open subspace of <math>X</math>" are used to mean that <math>(S,\tau_S)</math> is an open subspace of <math>(X,\tau)</math>, in the sense used above; that is: (i) <math>S \in \tau</math>; and (ii) <math>S</math> is considered to be endowed with the subspace topology.