Disclaimer: By a zero-dimensional space I mean a topological space having a base of sets that are at the same time open and closed in it.
At my university, we discussed zero-dimensionality and examples of zero-dimensional spaces.
I am asking whether the following spaces are zero-dimensional and how to, in general, find this about a space.
I only know that a subspace of zero-dimensional space is zero-dimensional and that for any compact topological space $X$, the notions of zero-dimensionality and strong zero-dimensionality coincide. Also, if $X$ is a compact Hausdorff space, then $X$ iszero-dimensional if and only if it is totally disconnected.
But none of this seems to be very helpful, so I will appreciate any recommendations what to use.
Question: Are these space zero-dimensional?
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$\omega \times \mathbb{R}$
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$\omega \times \mathbb{R}^n$
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$\omega \times S$ ($S$ = the circle $S^1$)
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$\omega \times S_n$, $n > 1$
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$R^n, n > 1$ (higher-dimensional Euclidean spaces)
I think the 5. is clear – because $\mathbb{R}$ is not zero-dimensional, $\mathbb{R}^n$ is not zero-dimensional either (it is hereditary property).
For the rest – I am not really sure. If $\mathbb{R}$ is not zero-dimensional, does it imply anything about the product with other spaces?
Thank you for your advice or for any sources to read and find answers to this.
Best Answer
None of your spaces is zero-dimensional. $X\times Y$ is zero-dimensional iff both $X$ and $Y$ are (if both are non-empty). The factor spaces embed as subspaces in the product etc. And of course any connected space like $\Bbb R^n$, $S$ or $S_n$ is not zero-dimensional. The $\omega$ (as a discrete space) is, but that's not enough..