A subset $E \subset X$ of a topological space $X$ is dense if $\overline{E} = X$ where

$$ \overline{E} = \bigcap \lbrace C \subseteq X \mid C \text{ is closed and } E \subseteq C \rbrace $$

But in the cofinite topology closed sets are defined to be finite sets. So if $X$ is infinite and a subset $E$ is dense, then this would imply that $X$ (a infinite set) is the intersection of finite sets. Does this mean that $X$ endowed with the cofinite topology has no dense subsets?

  • 2
    $\begingroup$ I think there is an intersection missing in your definition of density. $\endgroup$ – PAM Mar 1 '15 at 18:28
  • $\begingroup$ @Pam Yes you're right thank you! $\endgroup$ – Kevin Johnson Mar 1 '15 at 18:28
  • $\begingroup$ No; see whacka’s answer. In fact every infinite subset of $X$ is dense. $\endgroup$ – Brian M. Scott Mar 1 '15 at 18:35

There's one thing you're missing: there is exactly one infinite closed set available.

(Also no finite $C$s can contain $E$ if it's infinite so no finite sets will be used in the $\bigcap$.)

  • $\begingroup$ Ahh you mean $X$? Which would make every infinite subset dense $\endgroup$ – Kevin Johnson Mar 1 '15 at 18:36
  • $\begingroup$ Exactly. ${}{}$ $\endgroup$ – whacka Mar 1 '15 at 18:43

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