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I know what's the definition of 'open relative'. I googled 'close relative', but i couldn't find a definition of it.

How come every metric space $X$ is close relative to $X$?

If $p$ is a limit point of $X$, there exists a neighborhood $N_r(p)$ and $q\in X$ such that $q\in N_r(p)$ and $q≠p$.

What kind of definition makes $p$, a limit point of $X$ is a point of some set, thus $X$ is closed relative to $X$?

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The phrase is "closed relative to", not "close relative", so that is probably why google was not turning up any information. – Zev Chonoles Jul 24 '12 at 8:07
up vote 3 down vote accepted

Given a topological space $A$, we can endow a subset $B\subseteq A$ with the subspace topology.

We say that a subset $C\subseteq B$ is "open relative to $B\,$" when $C$ is open in the subspace topology on $B$; that is, by definition, when there is some open subset $U$ of $A$ such that $C=U\cap B$.

The definition is identical for "closed", i.e. a subset $C\subseteq B$ is "closed relative to $B\,$" when it is closed in the subspace topology on $B$, which (by definition) is when there is a closed subset $D$ of $A$ such that $C=D\cap B$.

Thus, it is a trivial result that every topological space $A$ is closed relative to $A$, because $A$ must be closed in its topology (that is part of the definition of a topology) and therefore $A=A\cap A$ is the intersection of a closed subset of $A$ with $A$.

Any metric space $X$, with distance function $d$, can be given a topology where the open subsets of $X$ are precisely those subsets $Y\subseteq X$ with the property that, for any $p\in Y$, there is an $\epsilon>0$ such that $B_\epsilon(p)\subseteq Y$, where $$B_\epsilon(p)=\{q\in X\mid d(p,q)<\epsilon\}$$ is the open ball of radius $\epsilon$ centered at $p$. Thus, any metric space can be given the structure of a topological space.

Thus, we can simply apply the above result (which holds for all topological spaces) to a metric space, and obtain the statement that any metric space $X$ is closed relative to $X$.

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