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Inspired by the confusion in the comments on this question:

I always thought that the standard was to read $\subset$ as "is a strict subset of", and $\subseteq$ could mean proper or improper inclusion.

Was I wrong?

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This goes to the list of questions like: Is $0$ a natural number? Will a ring have a multiplicative identity? In other words, depends on the author, and the author is expected to make her/his convention clear at the earliest convenient occasion. These may or may not lead to religious wars... –  Jyrki Lahtonen Jul 8 '11 at 5:39
    
Right. I didn't know at first that this was that kind of an issue. Always thought there was a general consensus. –  Josh Chen Jul 8 '11 at 5:45

4 Answers 4

up vote 8 down vote accepted

Different people use different conventions. Some people use $\subset$ for proper subsets and $\subseteq$ for possible equality. Some people use $\subset$ for any subset and $\subsetneq$ for proper subsets. Some people use $\subset$ for everything, but explicitly say "strictly proper" in words when they feel it matters. I do not believe that there is a consensus for the meaning of $\subset$. My own personal advice is to use $\subseteq$ and $\subsetneq$ when you care to be precise, and $\subset$ when you are feeling lazy.

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Just to give an example of what I mean, you might write "Since $\{1\}\subset \{1,2\}\ldots$, but "Let $X\subseteq Y\ldots$" or "Let $X\subsetneq Y\ldots$" –  Aaron Jul 8 '11 at 6:14
    
When sets are explicitly defined as subsets (as in your example, or $X\setminus\{x\}\subset X$) it is not "feeling lazy", it is being explicit. To write "when feeling lazy" sounds like a bad advice for mathematics. –  Asaf Karagila Jul 8 '11 at 6:25
    
@Asaf You're right that the example isn't strictly "being lazy", but I wouldn't be opposed to "Let $X\subset Y\ldots$" when a construction happens to coincidentally work when $X=Y$ but you don't really care about that particular case (perhaps because you are going to specialize later). Phrasing it as "when you're feeling lazy" is probably bad form on my part, and to a certain extent one should always be as precise as possible, but there are situations where the slight ambiguity doesn't cause any harm. –  Aaron Jul 8 '11 at 6:39

This is a very troubling issue with notations - it might not be uniform.

In many places $\subset$ implies a proper subset, while $\subseteq$ implies a possibly improper subset. In books you will find the definition somewhere, but in questions here... you just have to "guess" the right definition from the context.

Personally I am always in favor of clarity (when possible), $\subseteq$ and $\subsetneq$ are my choice of symbols. One of my teachers even takes $\subseteqq$ and $\subsetneqq$ for even greater clarity.

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Did you ever seen $A\subset\subset B$ for the $\overline{A}\subseteq B$? I saw it only in the Russian literature and wonder if it is also used worldwide. –  Ilya Jul 8 '11 at 6:08
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@Gortaur: I have never this notation before. –  Asaf Karagila Jul 8 '11 at 6:15
    
thank you, the same I heard from other people in my university. –  Ilya Jul 8 '11 at 6:26
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@Gortaur: Yes, I have seen that notation used in point-set topology and complex analysis... I think there's actually a question on this site (or mathoverflow) about that. –  Jesse Madnick Jul 8 '11 at 7:01
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@Gortaur: the notation $A \Subset B$ or $A \subset\subset B$ is actually stronger than what you wrote (at least in most contexts that I've seen, which is in analysis). Most authors I am familiar with take that to mean $A$ is compactly included in $B$, in the sense that $\bar{A}$ (the closure of $A$) is a compact subset of $B$. Note that this definition is outside the realm of notations as pure sets as it depends on topology. (In certain situations, for example, if $B$ were a bounded open subset of $\mathbb{R}^d$, you can appeal to Heine-Borel and get $A\Subset B\iff \bar{A}\subset B$.) –  Willie Wong Jul 8 '11 at 9:24

That depends on the author, see here.

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My convention has always been $\subset$ is strict and that $\subseteq$ is nonstrict. This maintains parallelism with $<$ and $\le$.

I have seen $\subset\subset$ in the comments. I have seen it used in this context. Write $K\subset\subset G$, when $K$ is compact and $G$ is open.

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