# Talk:Function to Product Space is Continuous iff Composition with Projections are Continuous

Lord Farin makes the comment after the line:

Let $U = \operatorname{pr}_{i_1}^{-1} \left({U_{i_1}}\right) \cap \cdots \cap \operatorname{pr}_{i_n}^{-1} \left({U_{i_n}}\right)$ be an open set in the natural basis of the product topology for $Y$.

"Of course, this is valid for finite $I$, but either add that to the assumptions or adapt the proof."

I thought that for the moment, but then I noticed that the natural basis of the product topology specifically states that $I$ is finite.

So by my understanding, as it currently stands it's okay ...? --prime mover 03:26, 12 December 2011 (CST)

Sorry, jumped to conclusions. The finiteness of these expressions reminds me of what I encountered when doing my BSc. thesis on lattice theory, where also a finiteness condition was imposed. Comment removed. --Lord_Farin 07:46, 12 December 2011 (CST)
I've added a line to stress the point about finitude. --prime mover 13:54, 12 December 2011 (CST)

## Rename suggestion

Rename: what to? Please clarify. --prime mover (talk) 15:41, 3 May 2018 (EDT)

I have suggested merging this page with Continuous Mapping to Topological Product as a General Result. This will resolve the rename issue. --Leigh.Samphier (talk) 21:28, 29 December 2019 (EST)
Sounds like a plan. Feel free to implement it. --prime mover (talk) 01:31, 30 December 2019 (EST)