Metric spaces

A metric space is a nonempty set {X} along with a function {d:X\times X\rightarrow\mathbb R} such that

  • {d(x,y)\geqslant 0}, with equality iff {x=y}.
  • {d(x,y) = d(y,x)}
  • {d(x,z) \leqslant d(x,y) + d(y,z)}.

Given a point {x\in } and a positive number {\varepsilon}, the open ball centered at {x} with radius {\varepsilon} is

\displaystyle B(x,\varepsilon) = \{y\in X: d(x,y) < \varepsilon \},

and the closed ball centered at {x} with radius {\varepsilon} is

\displaystyle \overline{B(x,\varepsilon)} = \{y\in X: d(x,y) \leqslant\varepsilon \}.

A set {U\subset X} is said to be open if for each {x\in U}, there exists {\varepsilon_x>0} such that {B(x,\varepsilon_x)\subset U}. A set {F\subset X} is said to be closed if {F^c=\{x\in X: x\notin F\}} is open.

Theorem 1 An open ball is an open set.

Proof: Let {x\in X} and {\varepsilon>0}, and let {y\in B(x,\varepsilon)}. Put {\delta=\varepsilon - d(x,y)}. Then if {d(y,z)<\delta},

\displaystyle d(x,z) \leqslant d(x,y) + d(y,z) < d(x,y) + (\varepsilon - d(x,y)) = \varepsilon,

so that {B(x,\varepsilon)} is open. \Box

Theorem 2 If {X} is a metric space, then {\varnothing} and {X} are open, the union of any collection of open sets is open, and the intersection of finitely many closed sets is open.

Proof: It is clear that {\varnothing} is open, as there is no {x\in\varnothing}. Since {B(x,\varepsilon)\subset X} for any {x\in X, \varepsilon>0}, {X} is open.

Let {\{U_\alpha\}_{\alpha\in I}} be a collection of open sets in {X}. Put {U=\bigcup_{\alpha\in I}U_\alpha}. If {x\in U}, then {x\in U_\beta} for some {\beta\in I}. Since {U_\beta} is open, there exists {\varepsilon_x} such that {B(x,\varepsilon_x)\subset U_\beta\subset U}. Hence {U} is open.

Let {U_1, U_2} be open sets in {X}. If {x\in U_1,U_2} then there exist {\varepsilon_1, \varepsilon_2} such that {B(x,\varepsilon_1)\subset U_1} and {B(x,\varepsilon_2)\subset U_2}. Put {\varepsilon = \min\{\varepsilon_1, \varepsilon_2\}}, then {B(x,\varepsilon)\subset (U_1\cap U_2)}, so that {U_1\cap U_2} is closed. The general result follows from induction. \Box

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