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Some people would say that the vectors $v$ and $-v$ have different directions. One direction is the negative of the other; the vectors are not parallel, they are anti-parallel. Other people would say that $v$ and $-v$ have the same direction (and that they are parallel). These people would then say that $v$ and $-v$ have different "sense". The first group ...


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This is typically defined by $$f(t)=\begin{cases} (1-2t)x+2tz & t\in[0,.5] \\ (1-(2t-1))z+(2t-1)y & t\in(.5,1] \end{cases}$$ That is, it's just the straight line from $x$ to $z$ followed by the straight line from $z$ to $y$. (Source: Munkres did this in lecture.)


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The two statements given in your question are precisely $$f(x) \in S \qquad \forall x\in S$$ i.e. $f: S\to S$ is a function and $$\forall y\in S \exists x\in S: f(x) = y$$ i.e. $f(S) = S$ or $f$ is surjective. If your notation reflects what you want to call "completely closed", it is nothing other than a surjective function from a set $S$ to itself.


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This order is sometimes called the shortlex order: length first, and lexicographic order for strings of the same length.



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