What definition of 'nowhere dense' is most useful for proofs in Topology? I know A is nowhere dense if every open set contains a point not in A.
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What is a "most useful" definition? As far as I can imagine, definitions are either correct or wrong. – Dominique Nov 16 '23 at 09:53
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1That is not at all the definition of "nowhere dense". With your definition $\mathbb{Q}$ would be nowhere dense in $\mathbb{R}$. – Captain Lama Nov 16 '23 at 09:54
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For various definitions of "nowhere dense" (and "empty interior") in a metric space setting, see my answer to Is saying a set is nowhere dense the same as saying a set has no interior? As to which (for metric spaces) is most useful, this depends on the context. For example, the version "For each $x \in X,$ each ball $B(x,\epsilon)$ contains at least one ball all of whose points belong to the complement of $X$" is the most useful when one is working with porous and $\sigma$-porous sets. – Dave L. Renfro Nov 16 '23 at 11:29
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Usually in these cases, there are a few options for which definition to pick. Whichever you pick, the rest will be proven as theorems relatively early after that. After those theorems have been proven, which is the definition and which are the theorems is more or less irrelevant, you use the one that suits your current need the best.
In this case, talking about "nowhere dense", we do have several options. For instance, a subset $A\subseteq X$ of a topological space is nowhere sense iff either of the following hold
- the closure of $A$ has empty interior
- for any non-empty open set $U\subseteq X$, there is a non-empty open subset $V\subseteq U$ that is disjoint from $A$
Any topology textbook author and any topology lecturer will have to pick one of those (or maybe some other option) as their definition, and prove as a theorem that the other option also characterises the exact same property. Once that's done, it doesn't really matter any more which one is chosen as "the definition". You can use which one you want to use depending on the problem you're facing, your mood, the day of the week, and so on.
So as long as the different possible definitions agree, the only real difference is which one is easiest to use as springboard to prove the rest.
Arthur
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