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Let $f : X → Y$ be a map of topological spaces. Show that $f$ is continuous if and only if $f\left(\bar{A}\right) \subset \overline{f(A)}$ for all $A \subset X$.

Seems to me that $\subset$ should be $\subseteq$. Taking example $X = Y = \mathbb{R}$ with usual topology, $f(x) = x$ and $A = [0,1]$ gives $f(\bar{A}) = \overline{f(A)}$. Have I missed something?

user27182
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    Depending on the authors, $\subset$ could mean $\subseteq$ – md5 Apr 19 '20 at 17:38
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    In higher mathematics it is very commonly used $;\subset;$ to denote weak contention of sets. We mathematicians are lazy people when it comes to write down stuff... – DonAntonio Apr 19 '20 at 17:38
  • Thanks to both of you.

    Do you think it should be $\subseteq$ in this case? (In meaning, not notation)

     – user27182 Apr 19 '20 at 17:38
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    @user27182 It should be, but many mathematicians (I included) use $\subset$ for what you mean by $\subseteq$ – Maximilian Janisch Apr 19 '20 at 17:39
  • @user27182 It is weak contention, as your example shows – DonAntonio Apr 19 '20 at 17:39
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    Add me as another vote for never writing $\subseteq$ (including 40 years of teaching and authoring 4 textbooks). If I want to be explicit about not allowing equality, I write $\subsetneq$. This way there is no doubt. – Ted Shifrin Apr 19 '20 at 17:42
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    After some 40 years of teaching and doing mathematics I vastly prefer to avoid $\subset$ altogether and use $\subseteq$ and $\subsetneqq$; it is simply a fact of life that $\subset$ is ambiguous, and I simply won’t use it. Unfortunately, many otherwise good writers can’t be bothered to make the minor effort required to avoid the potential ambiguity. – Brian M. Scott Apr 19 '20 at 17:47
  • I used to use $\subset$ only, but my field mostly involves combinatorics where it often matters. Someone called me out on it after a talk I gave at a combinatorics conference, where he pointed out that it was not topology. – Matt Samuel Apr 19 '20 at 17:58
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    @Brian M. Scott: Same with me. I almost never need to use a "proper subset" symbol, so the possible ugliness of $\subsetneqq$ is not much of an issue anyway. Note the profusion of $\subseteq$'s here and here. Same with $\log$ and $\ln,$ and other similar potential ambiguities whose usage to me seem akin to trying to win the battle while losing the war. See also this answer. – Dave L. Renfro Apr 19 '20 at 18:07

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The topology book I'm most familiar with, Munkres, spells out at the beginning that $\subset$ does not necessarily indicate proper containment because in topology it is so unusual for it to matter whether or not the containment is proper that they'd rather do this and instead write $\subsetneq$ for proper containment.

Matt Samuel
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