Espaço completamente regular

Em topologia, um espaço topológico é completamente regular se um conjunto fechado e um ponto externo a este conjunto podem ser separados por uma função contínua; ou, mais precisamente:[1]

Para todo conjunto fechado A não vazio [Nota 1] e todo ponto p, , existe uma função contínua f com contradomínio no intervalo [0, 1] tal que f(A) = { 1 } [Nota 2] e f(p) = 0.

Alguns textos incluem na definição de completamente regular que tenha a propriedade acima e também a propriedade T1, ou seja, que dois pontos p e q possam ser separados por abertos A e B (não necessariamente disjuntos) em que cada ponto pertence a um aberto mas não ao outro.

Um espaço T3 1/2 é definido, dependendo do livro consultado, como um espaço completamente regular, ou como um espaço completamente regular e T1.

Relação com os demais axiomas de separação

Ver artigo principal: Axiomas de separação

Notas e referências

Notas

  1. ↑ A literatura costuma negligenciar a restrição de A não ser vazio, o que é um erro, pois
  2. ↑ Novamente, a literatura costuma ser negligente com a notação, escrevendo f(A) = 1

Referências

  1. ↑ a b c University of Toronto, Department of Mathematics, Chapter 3: Separation Axioms [https://web.archive.org/web/20140407070227/http://www.math.toronto.edu/~mat1300/oldnotes/separation.pdf Arquivado em 7 de abril de 2014, no Wayback Machine. [em linha]]

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