# A metric space from closed sets

From Mendelson's Introduction to Topology:

Let $(X,d)$ be a metric space. ... [Then] $X$ is open.

But consider the closed interval $[a,b]$ over the real numbers. Presumably there is a distance function $d$ such that $([a,b],d)$ is a metric space. But $[a,b]$ is not an open set. What, then, have I missed?

To expand upon my question, here is my rough counterproof.

An open set is a neighborhood of each of its points. Consider the metric space $(X,d)$, with $X = [a,b]$ and $d(x,y) = \sqrt{x^2+y^2}$. If $X$ is open, then $X$ must be a neighborhood of $a$, which means there must be an open ball $B(a;\delta)\subset X$ for some $\delta > 0$. Yet if $\delta > 0$, then there is some $c \in B(a;\delta)$ such that $c < a$. Therefore $c \notin X$; a contradiction.

What is the flaw in my logic here?

• $[a,b]$ is open in itself. Any metric space is its own universe. By definition, any open ball is contained in the universe. – Prahlad Vaidyanathan Sep 22 '13 at 14:55
• $[a,b]$ is not open in $\mathbb R$, but it is open in $[a,b]$ itself. – Santiago Canez Sep 22 '13 at 14:57
• Think about this; $R$ is not open in $R^2$, but it is open in $R$. open/closed property is very dependent on the underlying space. (anyway how can I write the real number set R? mathds doesnt work... $\mathds{R}$) – dust05 Sep 22 '13 at 15:01
• Thanks all for your comments - I have expanded my question to clarify this. @dust05: \mathbb{R} – Doubt Sep 22 '13 at 15:17
• I think the natural definition of the open ball in $X$ is defined as $B(x;\delta)\cap X$ in this case. – dust05 Sep 22 '13 at 15:22

Or reread the definiion of open set exactly: If $(X,d)$ is a metric space, then for $x\in X$ and $r\in \mathbb R_{>0}$ the set $B_r(x):=\{\,y\color{red}{\in X}\mid d(x,y)<r\,\}$ is called open $r$-ball around $x$ and $U\subseteq X$ is called open iff for each $x\in U$, there exists some $r>0$ with $B_r(x)\subseteq U$. For $X=[0,1]$ and the usual metric note that these definition make e.g. $B_{\frac12}(\frac13)=[0,\frac56)$ an open set.