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Homemetric space
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metric space
A metric space is a set $X$ together with a real valued function $d:X\times X\longrightarrow\mathbb{R}$ (called a metric, or sometimes a distance function) such that, for every $x,y,z\in X$,

$d(x,y)\geq 0$, with equality^{1}^{1}This condition can be replaced with the weaker statement $d(x,y)=0\iff x=y$ without affecting the definition. if and only if $x=y$

$d(x,y)=d(y,x)$

$d(x,z)\leq d(x,y)+d(y,z)$
For $x\in X$ and $\varepsilon\in\mathbb{R}$ with $\varepsilon>0$, the open ball around $x$ of radius $\varepsilon$ is the set $B_{\varepsilon}(x):=\{y\in X\mid d(x,y)<\varepsilon\}$. An open set in $X$ is a set which equals an arbitrary (possibly empty) union of open balls in $X$, and $X$ together with these open sets forms a Hausdorff topological space. The topology on $X$ formed by these open sets is called the metric topology, and in fact the open sets form a basis for this topology (proof).
Similarly, the set $\bar{B}_{\varepsilon}(x):=\{y\in X\mid d(x,y)\leq\varepsilon\}$ is called a closed ball around $x$ of radius $\varepsilon$. Every closed ball is a closed subset of $X$ in the metric topology.
The prototype example of a metric space is $\mathbb{R}$ itself, with the metric defined by $d(x,y):=xy$. More generally, any normed vector space has an underlying metric space structure; when the vector space is finite dimensional, the resulting metric space is isomorphic to Euclidean space.
References
 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Mathematics Subject Classification
54E35 no label found8200 no label found8300 no label found8100 no label found Forums
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