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Getting into topology this semester and have been studying and doing some exercises and I was wondering if a Hausdorff space i.e: $ {\displaystyle \forall \,x,y\in X\,|\,x\neq y\,\exists \,U,V{\text{ open sets of }}x,y\,|\,U\cap V=\varnothing }$ Could eventually be a connected space ie: X is connected, that is, it cannot be divided into two disjoint non-empty open sets. ,even if at first to me they seem to counter each other there are claims they exists, but I can't find any examples and I just found this very poor wiki link. Of course if anyone has a proof I would more than appreciate it or any literature that could give me an answer

https://topospaces.subwiki.org/wiki/Connected_Hausdorff_space#:~:text=There%20do%20exist%20countably%20infinite%20connected%20Hausdorff%20spaces.

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    $\begingroup$ Every metric space is Hausdorff (for $U$ and $V$ you can take open balls of sufficiently small radius) and many are connected, for example $\mathbb{R}$. $\endgroup$ Commented Jun 14 at 7:28
  • $\begingroup$ Thank you guys, much more clear now $\endgroup$ Commented Jun 14 at 7:31
  • $\begingroup$ An example which is not metric space is $\Bbb R$ with $K$-topology. $\endgroup$
    – Bob Dobbs
    Commented Jun 14 at 7:56

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Being connected and Hausdorff are two independent properties:

  1. Connected + Hausdorff: the real line $\mathbb{R}$ with the standard Euclidean topology. Or a single point $\{x\}$ with discrete topology $\tau=\{\emptyset, \{x\}\}$.
  2. Disconnected + Hausdorff: two points $\{x,y\}$ with discrete topology $\tau=\{\emptyset, \{x\}, \{y\}, \{x,y\}\}$.
  3. Connected + non-Hausdorff: two points $\{x,y\}$ with Sierpiński topology $\tau=\{\emptyset, \{x\}, \{x,y\}\}$. It is similar to discrete, except $\{y\}$ is not open.
  4. Disconnected + non-Hausdorff: two copies (disjoint union) of Sierpiński space: $\{x,y\}\sqcup\{x', y'\}$. Or disjoint union of Sierpiński space with any other non-empty space.

Btw, disconnected spaces always arise as disjoint union of some of its subspaces.

Roughly speaking being Hausdorff is about relationship between points, you can think of it as kind of local property. While being connected is a global property. Except local connectedness is another different concept.

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