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?
$\endgroup$ 24 Answers
$\begingroup$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.
$\endgroup$ 3 $\begingroup$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.
$\endgroup$ 10 $\begingroup$That depends on the author, see here.
$\endgroup$ $\begingroup$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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