Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $X$?

share|improve this question
4  
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
add comment

1 Answer 1

up vote 2 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 $X$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.