Espacio de Sierpinski

No debe confundirse con Conjunto de Sierpinski.

En matemáticas, el espacio de Sierpiński (o el conjunto de dos puntos conectados) es un espacio topológico finito formado por dos puntos, de los cuales sólo uno es cerrado. Es el ejemplo más pequeño de espacio topológico que no sea el trivial ni el discreto. Su nombre es en honor a Wacław Sierpiński.

Definición

El espacio de Sierpiński es el conjunto $S=\{0,1\}$ con la topología ${\mathcal {T}}_{S}$ siguiente:

{\mathcal {T}}_{S}=\{\emptyset ,\{0\},S\}

Propiedades

Propiedades básicas del espacio de Sierpinski:^[1]

Los únicos conjuntos abiertos son $\emptyset$ , $\{0\}$ y $S$ .
Los únicos conjuntos cerrados son $\emptyset$ , $\{1\}$ y $S$ .
La clausura de $\{1\}$ es $\{1\}$ y la de $\{0\}$ es $S$ .
Es un espacio de Kolmogórov ( $T_{0}$ ).
No es un espacio de Fréchet ( $T_{1}$ ).
No es un espacio de Hausdorff ( $T_{2}$ ).
No es un espacio $T_{n}$ con $n>1$ .
Es un espacio compacto.
Es 1AN y 2AN.
Toda sucesión en $S$ converge a 0.
Si una sucesión en $S$ converge a 1, entonces tiene un número finito de términos iguales a 0.
Es un espacio no metrizable.
Es un espacio conexo.

Véase también

Bibliografía

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 edición), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446 .

Referencias

↑ Sapiña, R. «Espacio de Sierpinski». Problemas y Ecuaciones. ISSN 2659-9899. Consultado el 20 de septiembre de 2019.

Datos: Q2119470

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