Ponto isolado

Em topologia, um ponto $x$ de um espaço topológico $X$ é dito um ponto isolado de um subconjunto $S\subseteq X$ se $x\in S$ e existe em $X$ uma vizinhança perfurada de $x$ que não contém nenhum ponto de $S$ .

Em particular, em um espaço métrico, um ponto $x$ é dito isolado se existe $\varepsilon >0$ tal que $x$ é o único ponto de $S$ no intervalo $(x-\varepsilon ,x+\varepsilon )$ , ou seja, se existe uma bola em torno de $x$ que não contém nenhum ponto de $S$ ^[1]. Equivalentemente, um ponto $x\in S$ é dito isolado se e somente se ele não é um ponto de acumulação de $S$ .

Um conjunto cujos elementos são todos pontos isolados é dito um conjunto discreto. Um conjunto que não contém pontos isolados é dito denso em si mesmo. Um conjunto fechado que não contém pontos isolados é chamado de conjunto perfeito.

Exemplos

No conjunto $S=\{0\}\cup [1,2]$ , o ponto $0$ é um ponto isolado.
O conjunto $S=\left\{{\frac {1}{2}},{\frac {2}{3}},{\frac {3}{4}},{\frac {4}{5}},\ldots ,{\frac {n}{n+1}},\ldots \right\}$ é discreto, já que seus pontos são todos isolados, e seu único ponto de acumulação é o número $1$ , que não pertence ao conjunto^[2].
O conjunto $\mathbb {N} =\{0,1,2,3,\ldots \}$ dos números naturais é um conjunto discreto.

Referências

↑ LIMA, Elon Lages (2004). Análise Real Volume 1, Funções de Uma Variável Real Décima ed. [S.l.]: IMPA. p. 52. ISBN 978-85-244-0048-3
↑ ÁVILA, Geraldo (1999). Introdução à Análise Matemática Segunda ed. [S.l.]: Edgard Blücher. p. 76. ISBN 978-85-212-0168-7

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