Confusion in symbol of subset.

I'm afraid the usage varies by author.

Some of the symbols are unambiguous: '$\subsetneq$' always means a proper subset, and '$\subseteq$' means a subset which could either be proper or not.

But the symbol '$\subset$' is ambiguous. For some, it always means a proper subset (identical to '$\subsetneq$'. For others, it just means a subset without regard to properness (identical to '$\subseteq$').

My personal preference is to use '$\subset$' to just mean any subset, proper or not. When I specifically want to exclude the possibility of equality, I explicity indicate this using '$\subsetneq$'.

As @MPW pointed out, $\subset$ is ambiguous. My preference, therefore, is not to use this symbol at all. I use $\subseteq$ for "subset" and $\subsetneq$ for "proper subset". So you won't encounter any ambiguities in my papers; as for other people's papers, good luck.