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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.

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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
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 $X$.

For a demonstration of this let $X=\mathbb{R}^{2}$ and $x_{0}=0$. Assuming that $X$ has the discrete topology, then $A$ is closed, not dense and disconnected. But if we assume that $X$ has the Euclidean topology, then $A$ is not closed, dense and connected.

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$X \setminus \{x_0\}$ 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 $\mathbb{R}^2$).

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