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.

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