Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Which properties hold for the following set?
Open, connected, dense, closed
$A$=$X\setminus\{x_0\}$ where $X$ is an arbitrary Hausdorff topological space and $x_0\in X$.

Since $X$ is hausdorff it is $T_1$hence $A$ is open. But how can I able to verify the other properties.

share|improve this question
3  
You could profit from working out a few examples, for instance $X=\Bbb R, x_0=0$, $X=\lbrace 0,1,2,\rbrace, x_0=0$ –  Olivier Bégassat Jan 2 '13 at 13:06
add comment

2 Answers

up vote 3 down vote accepted

As you have noticed, $T_{1}$ implies that $A$ is always open. For rest of the properties there is no definite answer and they depend on the topology of $X$.

Take $X=\mathbb{R}^{2}$ and $x_{0}=0$. Assuming that $X$ has the discrete topology, then $A$ is closed, not dense and disconnected. Assuming that $X$ has the Euclidean topology, then $A$ is not closed, dense and connected.

share|improve this answer
add comment

$X \setminus \{x_0\}$ is dense iff $x_0$ is not an isolated point. It is closed iff $x_0$ is an isolated point. (which makes sense as a closed and dense subset of $X$ must equal $X$ by definition, so it cannot be both)

As you said, it is always open (in a $T_1$ space). For connected $X$, it is disconnected iff $x_0$ is a so-called non-cutpoint. (contrast $\mathbb{R}$ vs $\mathbb{R}^2$).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.