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Question: If $T$ and $T'$ are topologies on $X$ and $T'$ is strictly finer than $T$, what can you say about the corresponding subspace topologies on the subset $Y$ of $X$?

I can never really know what more is required for questions like this. I would simply answer that $T'$ has fewer elements in it than $T$ and leave it at that? Figured saying something like $Y \subset X$ would be too obvious?

What more must I say?

Siyanda
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2 Answers2

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The inherited topologies can be the same. So nothing more than that can be said in general.

Let $X$ have topologies $T,T'$ with $T'$ strictly finer than $T$. Denote these by $X_T, X_{T'}$. Let $Y$ be a non-empty subset of $X_T$. Then consider $Z_1=Y\times X_T$ and $Z_2=Y\times X_{T'}$, with the product topologies on both spaces. These are the same spaces as sets, and $Z_2$ has a strictly finer topology than $Z_1$. But the subsets $Y\times\{1\}$ have the same inherited topology. We could also take $Y$ to be a non-empty subset of $X_{T'}$. We don't even really need it to be related to $X$ and $T,T'$ at all, actually.

zibadawa timmy
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    Nice general construction. For a simpler example, look at the usual and discrete topologies on $\Bbb R$: both induce the discrete topology on $\Bbb Z$. – Brian M. Scott Nov 07 '13 at 10:29
  • I am confused about one part of this, when you say that $Y$ is a non empty subset of $X_T$, did you mean it is a subspace, or its a subset of the topology? Else I don't understand how if Y is not a topological space, how do we induce the product topology on $Y\times X_T$? – H_B Jan 15 '15 at 10:23
  • @H_B here subsets of a topological space inherit the subspace topology. – zibadawa timmy Jan 15 '15 at 21:21
  • OK I just did not understand if you meant Y was a set or a space. Thanks for getting back to me on a question that is over a year old. – H_B Jan 15 '15 at 22:38
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The 'finer' relation still holds for the subspace topologies.

Henry
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