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To my understanding, if we have a set $U$ and a $A\subset U$ then $A$ is relatively open to $U$ if $\exists$ and open set $\Omega\subset U$ s.t $A=\Omega \cap U$. But also a subset of $U$ is open $\iff$ it is relatively open so would it isn't wrong to say that $\Omega = A$ so why do we need to define the idea of openness relative to a set?

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  • $\begingroup$ $U$ is open if it is relatively open to what set? The thing about relative openness is says something about how we are considering the set in question. $\endgroup$ – Aaron Zolotor Oct 21 '18 at 17:34
  • $\begingroup$ I missed out part of the question! Thanks for pointing that out! $\endgroup$ – BigWig Oct 21 '18 at 17:36
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Take for example $\mathbb{R}$ with the standard topology. $A = [0, 1)$ is not an open, but if you take $U = [0, 2]$, you see that $A$ is relatively open with respect to $U$, because $[0, 1) = [0, 2] \cap (-1, 1)$, which is the intersection of $U$ and an open subset of $\mathbb{R}$.

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  • $\begingroup$ Oh!! so $\Omega$ itself doesn't need to be a subset of $U$? Would it also be that my assumption $A$ is open iff $A$ is relatively open in $U$ is false then? $\endgroup$ – BigWig Oct 21 '18 at 17:41
  • $\begingroup$ No, $\Omega$ is a subset of $\mathbb{R}$, in this case. More generally, if you had $U \subseteq V$ (both topological spaces), then $\Omega$ is a subset of $V$. This is known as induced or subspace topology. Yes, it's false, the example I showed above gives you a set not open in $\mathbb{R}$, but that's open in the subspace topology of $[0, 2]. $\endgroup$ – user512979 Oct 21 '18 at 17:47
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Consider $ \mathbb R$ with usual topology and $(0,1]$ with subspace topology wrt $\mathbb R$

Note that $(1/2,1]$ is open in $(0,1]$ but not open in $\mathbb R$

i.e. $(1/2,1]$ is relatively open in $(0,1]$

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  • $\begingroup$ So would it be that in your example $\Omega$ = A = (1/2, 1] in this case? $\endgroup$ – BigWig Oct 21 '18 at 17:40
  • $\begingroup$ Actually you have phrased your question wrongly. Relative openness makes sense when you want to talk about openness of set in some subset of topological space. $\endgroup$ – Mayuresh L Oct 21 '18 at 17:44

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