# Difference in between full and also shut set

What is the distinction in between a full statistics room and also a shut set?

Can a set be shut yet not finish?

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2019-05-19 00:01:18
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A statistics room is full if every Cauchy series merges. A part $F$ of a statistics room $X$ is shut if $F$ has every one of its restriction factors ; this can be identified by claiming that if a series in $F$ merges to a factor $x$ in $X$, after that $x$ have to remain in $F$. It additionally makes good sense to ask whether a part of $X$ is full, due to the fact that every part of a statistics room is a statistics room with the limited statistics.
It ends up that a full subspace has to be shut, which basically arises from the reality that convergent series are Cauchy series. Nonetheless, shut subspaces need not be full. For an unimportant instance, start with any kind of insufficient statistics room, like the sensible numbers $\mathbb{Q}$ with the common outright value range. Like every statistics room, $\mathbb{Q}$ is enclosed itself, so there you have a part that is shut yet not finish. If taking the entire room feels like disloyalty, simply take the rationals in $[0,1]$, which will certainly be enclosed $\mathbb{Q}$ yet not finish.
If $X$ is a full statistics room, after that a part of $X$ is shut if and also just if it is full.