# Continuity of a map of a topological space to a pro-topological space

Let $(X_i)$ be a projective system of topological spaces. Let $X$ be the projective limit of $X_i$.

Let $G$ be a topological space.

What does it mean for $G\to X$ to be continuous?

My guess is that there are continuous maps $G\to X_i$ which fit into some commutative diagrams.

Is that correct?

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I'm not sure what you want to know: do you want to know how to construct the projective limit of a system of topological spaces or do you want to know how to verify its universal property for a specific construction? Anyway: You do have continuous maps $X \to X_i$ by the very definition of the projective limit and by composition you get maps $G \to X_i$ compatible with the projective system, so it is necessary that those maps are continuous. On the other hand, the universal property ensures that there is a unique map $G \to X$ induced from the maps $G \to X_i$; this yields sufficiency. –  t.b. Apr 27 '12 at 10:51
yeah that's what i wanted to be sure of. –  Harry Apr 27 '12 at 15:46