Dropping excess symbols when writing mathematics is a common practice (when was the last time you read a proof that began: let (R,+,*) be a ring?). But when we try to let notation do the thinking for us this practice may lead us into error. For example:

A topological space, by definition, consists of a set *X* and a topology *T* on *X*, and so technically should be denoted (*X*,*T*). Writing that all the time, however, wastes precious paper, so such a space is usually denoted *X*.

The trouble comes in when we talk about continuous maps of topological spaces. For a map *f* between topological spaces to be continuous, it must be that the preimage of a set open in the codomain's topological space is open in the domain's topological space. Let *X* be a set, and let *T*, *T'* be distinct topologies on *X*. Then the identity map written in the ordinary fashion (id:*X*->*X*, where id(x)=x) certainly looks continuous---after all, it doesn't change anything; it's the identity map!

But when the topologies are distinct, the identity map *does* change things. For suppose that *X* contains at least two points, *T* is the indiscrete topology, and *T'* is the discrete topology. The singleton {x} is open in the discrete topology, but its inverse image {x} is not open in the indiscrete topology. So the identity map is not always continuous.