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I very often have to write something like:

$\exists U,V\subseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like:

$\exists U,V\mathop{\subseteq}_\text{open}M$ is there a common short hand though?

Example of a short hand that exists $\newcommand{\bigudot}{\mathop{\bigcup\mkern-14mu\cdot\mkern5mu}}$ $\newcommand{\udot}{\cup\mkern-11.5mu\cdot\mkern5mu}$
It is established already that $\bigudot$ means "union of sets that are pairwise disjoint", so if I write:

$\forall A\exists\mathcal{A}:A\subseteq\bigudot\mathcal{A}$ - or something - it is clear from the context that $\mathcal{A}$ is a family of pairwise disjoint sets

So my question is this:

Is there a notation for this already, like perhaps a $\subset$ with a dot in to mean "open subset"

Alec Teal
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    Open is a relative notion, it is not a property of the set. I guess that is why the notation (that I have seen) uses what gives it that property. $U\in\tau$ where $\tau$ is the topology. – Karanko Apr 06 '15 at 00:21
  • I am not sure how standard it is, but I have seen it written $U^{\text{open}}\subset M$. – N. Owad Apr 06 '15 at 00:21
  • @Karanko if $M$ above was a topological space, the notion of open becomes clear. What you're saying is "well not even writing 'open' is sufficient, because open is a relative property" - which is absurd – Alec Teal Apr 06 '15 at 00:22
  • Absurd is your syllogism that my comment implies that I am saying that "is open" is not sufficient. – Karanko Apr 06 '15 at 00:29
  • Oh you crafty person @Karanko - didn't see the edit. – Alec Teal Apr 06 '15 at 00:31
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    I suspect that whatever your doing in your notes is fine, but elsewhere, there's not all that common a notation - and in most elementary contexts "Let $U$ and $V$ be open subsets in $M$" would be much clearer than any symbol you could introduce (but in other contexts, references to the topology as a pair $(S,\tau)$ might be preferred). – Milo Brandt Apr 06 '15 at 00:32
  • @Meelo that's my experience! It's also why I asked, I hate slipping \text{ open } somewhere into my statements, but equally I don't want to "invent" a symbol for it - thanks for confirming (or answering with "no" ) – Alec Teal Apr 06 '15 at 00:34
  • I've seen some notes do $U = {}^{\circ}U\subset M$, but I can't say that it really helps. I would take into account though, that if you "edit" the subset symbol to include openness, it will stop behaving as a subset symbol does with sets. Notably, the transitive property will fail (starting with the fact that a chain of such symbols - denoting openness - would have an ambiguous meaning to begin with). I'm not talking from a position of great experience here either, but I believe that modifications to notation are to behave similar to their original counterparts. – GPerez Apr 06 '15 at 01:13
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    I've seen/used $X^\circ$ to represent the interior of $X$. Of course, $X$ is open iff $X = X^\circ$. I think this was Kuratowski's notation. – nomen Nov 01 '19 at 23:34

2 Answers2

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Remember that a topological space $X$ is really a pair $(X,\mathcal{T})$ where $\mathcal{T}$ is a collection of "open" subsets of $X$. We often drop this cumbersome notation. But if you write, $U\in \mathcal{T}$, then it is clear you are talking about open sets.

For example, you can define continuity $f:X\to Y$ by saying, $$ \forall U\in \mathcal{T}_Y, ~ f^{-1}(U) \in \mathcal{T}_X $$

Nicolas Bourbaki
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  • I'm asking for a subset notation. Which topology? The subset topology or open WRT the parent topology? Also writing $U\subset V\wedge U\in\mathcal{J}$ isn't really a short hand symbol – Alec Teal Apr 06 '15 at 00:28
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    Furthermore! We rarely explicitly mention the topology - even when studying topology the $(X,\mathcal{J})$ notation was quickly dropped. IME anyway – Alec Teal Apr 06 '15 at 00:30
  • If $A$ is an open subset of $B$, where $B$ is a subset of $X$ (topological space) then you can simply write $A\in \mathcal{T}_B$, and it becomes clear that $\mathcal{T}_B$ is referring to the topology on $B$ and not on $X$. – Nicolas Bourbaki Apr 06 '15 at 00:35
  • See that's missing the point, and it's not really a "convention", I would use $\mathcal{J}_X$ for a topology on $X$ say, you would use T - see the cup with a dot in it example which does away with the words "pairwise disjoint" - I'm also reluctant to mention a specific topology because it doesn't matter, for the same reason the neighbourhood part is dropped when defining germs - the neighbourhood doesn't matter. – Alec Teal Apr 06 '15 at 00:38
  • I never seen your dot-union notation before to mean disjoint union. As far as I know, there is no universal notation for disjoint union. Most people use a square-shaped U but every author simply just explains that it is a disjoint union. The same is true with "open". There is no universal notation for open, whenever topological arguments become delicate authors just spell out which set is open where. I think you are over-thinking this notation, in the end it does not matter so much. – Nicolas Bourbaki Apr 06 '15 at 00:41
  • There wont be a universal notation for anything, the circle with the + is sometimes used for disjoint union (especially in older texts) - this question was about establishing if one exists, rather than commenting "in a topology you can express via membership" - which anyone who's ever defined a topology knows (and is patronising) - you went this way. – Alec Teal Apr 06 '15 at 00:44
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I use $U \subseteq_{op}X$, but I writing "${op}$" underneath the subset symbol. Looks nice, quick to write, gets the message across. Three traits of good notation.

Spencer Kraisler
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  • I might add that the letters $U$, $V$, and $W$ are oft used to denote open subsets, while $Z$ or maybe $C$ often denotes a closed subset, which really helps clarify things in context as well. – Keeley Hoek Dec 06 '20 at 13:09