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Let $X$ and $X'$ denote a single set in the topologies $\mathscr T$ and $\mathscr T'$ respectively; let $Y$ and $Y'$ denote a single set in the topologies $\mathscr U$ and $\mathscr U'$ respectively. Assume these sets are non-empty.

a) Show that if $\mathscr T'\supseteq\mathscr T$ and $\mathscr U'\supseteq\mathscr U$, then the product toplogy on $X'\times Y'$ is finer than the topology on $X\times Y$.

Having trouble making sense of this problem. How can I compare the topologies if they are on different sets.?

Austin Mohr
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Tim Raczkowski
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    They aren't on different sets. As sets X=X', and Y=Y' – Seth Feb 21 '15 at 20:05
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    I think maybe $X=X'$ and $Y=Y'$ in terms of sets. The notation is to distinguish different topological spaces. –  Feb 21 '15 at 20:06
  • Yes, thank you Seth and Starlight. I was misunderstanding the notation. – Tim Raczkowski Feb 21 '15 at 20:07
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    I became confused on this problem as well since he does make it sound like there is some topological space (Z,t) and (Z,t') where X,X',Y,Y' are elements of the topology. I would think to then talk about their product topology you would first have to define topologies on X,X',Y,Y', perhaps with the subspace topology or something like that. – user1236 Feb 29 '16 at 20:15

1 Answers1

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The notation is a bit confusing. As a topological space is just a pair $(X, \mathcal{T})$ it would have been better stated, I think, as follows:

Suppose we have non-empty topological spaces $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ and another pair of non-empty topological spaces $(Y,\mathcal{U})$ and $(Y,\mathcal{U}')$.

As an auxiliary notation, denote the product topology of two topologies $\mathcal{T}$ and $\mathcal{U}$ by $\mathcal{T} \otimes \mathcal{U}$. As you probably know, this is the smallest topology on $X \times Y$ that contains the set $\{U \times V: U \in \mathcal{T}, V \in \mathcal{U} \}$.

If $\mathcal{T} \subseteq \mathcal{T}'$ and $\mathcal{U} \subseteq \mathcal{U}'$, then the product topological space $(X \times Y, \mathcal{T}' \otimes \mathcal{U}')$ has a finer topology than $(X \times Y, \mathcal{T} \otimes \mathcal{U})$, or more succinctly: $\mathcal{T} \otimes \mathcal{U} \subseteq \mathcal{T}' \otimes \mathcal{U}'$.

Henno Brandsma
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  • Thanks for the enlightening answer. I notice that the same notation use for the product topology is also used for the product of $\sigma$-algebras in Folland's Real Analysis. Are these notations standard? – Tim Raczkowski Feb 22 '15 at 16:44
  • I've seen it used mostly for $\sigma$-algebras. – Henno Brandsma Feb 22 '15 at 17:25