Proper dense open subset of X

general-topology

$X$ be a topological space and $U$ be a proper dense open subset of $X$.
Then pick the correct statement from the following:

  1. If $X$ is connected then $U$ is connected.
  2. If $X$ is compact then $U$ is compact.
  3. If $X\setminus U$ is compact then $X$ is compact.
  4. If $X$ is compact then $X\setminus U$ is compact.

Answer:

My approach is, while $U$ is open and dense subset therefore if $X$ is compact and $U$ is open then $X\setminus U$ is closed subset of X and also it is non-empty, hence $X\setminus U$ is compact.

So
option 4 is correct.But I don't know about other options. What to do?

Best Answer

  1. $X=[-1,1]$ (usual topology), is connected and its subspace $U=\mathbb{R}\setminus \{0\}$ is open and dense and disconnected. False.

  2. As 1., note that a compact subset of $[-1,1]$ is closed and $U$ is not.

  3. As 1. $X\setminus U=\{0\}$ is compact.

  4. True, as $X\setminus U$ is closed in $X$ ($U$ is open) and so compact too when $X$ is.

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