Homeomorphism between topological space and product space

general-topology connectedness product-space

Yes, if $I$ is an infinite set and $Y$ any topological space, then $X = Y^I$ is homeomorphic to $X\times X$. If $Y$ is connected, so is $X$. A related but different example is given by the $\ell^p(\mathbb{N})$ spaces.


Yes. Take any infinite set equipped with the trivial topology. A singleton also has this property.

For a more interesting example, see: Hilbert cube.

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