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I am reading Munkres's topology, and exercise 4 on p. 91 asks to prove the following:

Show that $\pi_1: X \times Y \rightarrow X$ and $\pi_2: X \times Y \rightarrow Y$ are open maps.

I can prove this if $\pi_n$ are projections, but my question is: what would be "weak" conditions on $\pi_n$ for this statement to hold? I feel that specifying the functional form such as $\pi_1(x,y)=x$ is quite strong and can be relaxed, since maps to the constituent spaces of the product are natural.

My motivation is that if I find a space that can be decomposed as a product space, whether there are good "tests" to check whether there are maps to the new "axes" spaces that are not simply projections, since if the map is open and surjective, it is a quotient map and the resulting quotient topologies may be different that that induced by the projection.

xletmjm
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  • You can find the solution explained in detail here. – eta.beta Dec 21 '24 at 11:34
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    No, because what you linked to assumes that these are projections – xletmjm Dec 21 '24 at 11:51
  • Without any restriction on a map $X \times Y \rightarrow X$, there will be most often a counter-example. For instance, take any continuous, not injective map $f: \mathbb R \rightarrow \mathbb R$. Then f is not open, hence $f \pi_1: \mathbb R \times \mathbb R \rightarrow \mathbb R$ is continuous, not open (where $\pi_1$ is the projection). – Ulli Dec 21 '24 at 12:56
  • @Ulli that the point of the question - what are some restrictions (weaker than being a projection) that guarantee openness of the map from a product space to the component spaces? clearly there are whole classes of maps (such as the ones you outlined) that cannot be open, but perhaps there is a class of functions that are open because of how the product space is constructed. For example $f: S^1 \times \mathbb{R} \rightarrow S^1$ such that $f(\theta, x) = \theta + \pi \mathrm{mod} 2 \pi$ is not a projection but is open map (as a composition of a projection and a bijection) – xletmjm Dec 22 '24 at 21:41

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