I am reading "Introduction to Topology" by Gamelin and Greene (2nd edition) and I am confused by what appears to be an inconsistency in notation for subsets and strict subsets.
The image below shows page 61 from this book. From the solid red highlighted expressions, it appears that $A \subset B$ means $A$ is a subset (not strict) of $B$. However, from the blue dashed highlighted expression, it appears that $A \subseteq B$ means $A$ is a subset (not strict) of $B$.
Is this a typo in the book, or have I misunderstood something?
Frustratingly, they don't define the symbols $\subset$ and $\subseteq$ anywhere.
UPDATE: Thanks to the helpful comments and answer, it's clear that there is a notational inconsistency and the authors mean "subset" (not "strict subset") with both $\subset$ and $\subseteq$ on pg 61.
But, is it safe to assume they always mean "subset" (and never have a use for "strict subset") through the entire book? I'm new to topology, and there are other places where I cannot figure out whether they mean "strict subset". Is it likely that the entire book would never have a use for "strict subset"?
$\subset$ and $\subseteq$ have the same meaning. Most authors prefer $\subset$ , but there are also authors using $\subseteq$ . It is just a matter of taste. However, I agree to you that one should not use both symbols in the same text.
Having a quick view into the book, I only found very few occurrences of $\subseteq$ . I would therefore regard these as "unintended" occurrences. Perhaps the authors (there are two of them!) had different preferences, or made typos in the manuscript, or $\subseteq$ was used erroneously in the print set created by the book publisher - but that is pure speculation.
Your example $\operatorname{int} S \subset S$ from p. 61 contrasts to $\operatorname{int} Y \subseteq Y$ on p. 3. This should say enough: The use is not standardized in the book.