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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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    Right. I didn't know at first that this was that kind of an issue. Always thought there was a general consensus.2011-07-08

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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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    @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.2011-07-08
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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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    @Gortaur: in all situations I've seen, the notation $A \Subset B$ means $A$ is a relatively compact subset of $B$.2011-07-08
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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.