54A05-CharacterizationOfConnectedCompactMetricSpaces.tex

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\pmtitle{characterization of connected compact metric spaces.}
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Let $(A,d)$ be a compact metric space. Then $A$ is connected  $ \Longleftrightarrow \forall x,y \in A, \; \forall \varepsilon >0\; \exists \; n \in \mathbb{N}, \; \text{and } p_1,...,p_n \in A, \; \text{such that } p_1=x, \; p_n=y \; \text{and } d(p_i,p_{i+1})<\varepsilon \;(i=1,...,n-1)$
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