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Is there a countable connected space $(X,\tau)$ such that for all $x\in X$ the space $X\setminus\{x\}$ is not connected any more with the induced subspace topology?

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    $\begingroup$ Can’t resist: The one point space satisfies this. $\endgroup$ – Thomas Rot Dec 25 '18 at 19:24
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    $\begingroup$ @Thomas: But the empty space is connected! $\endgroup$ – Fred Rohrer Dec 25 '18 at 20:15
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    $\begingroup$ @FredRohrer This is a convention, but usually, I would say no. For the same reason that 1 isn't prime. Otherwise you cannot say that a space uniquely decomposes into a disjoint union of connected spaces... $\endgroup$ – Najib Idrissi Dec 25 '18 at 21:32
  • $\begingroup$ @FredRohrer: It depends on the convention indeed. $\endgroup$ – Thomas Rot Dec 25 '18 at 21:35
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    $\begingroup$ @FredRohrer Your statement is also true if $\varnothing$ is not connected. I think there are many reasons to want it to not be connected: unique decomposition in connected components; $\hom(X,-)$ preserves coproducts if $X$ is connected; for a (path-)connected space, $\pi_0(X) = *$; a product is connected iff both factors are connected. (This isn't off the top of my head, I'm reading this.) Anyway, this is a bit tangential, and as you say we're certainly rehashing old arguments... $\endgroup$ – Najib Idrissi Dec 26 '18 at 8:54
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Let $\mathbb{R}$ be with its usual topology, and let $f:\mathbb{R}\to \mathbb{Z}$ defined by: $$ f(x)=\left\{\begin{matrix} 2k & x=2k,\text{ where } k\in\mathbb{Z}\\ 2k+1 & 2k<x<2k+2,\text{ where } k\in\mathbb{Z} \end{matrix}\right. $$ Let $X$ be $\mathbb{Z}$ with the quotient topology induced by $f$.

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    $\begingroup$ In other words it is the Alexandroff topology for the partial order$$\cdots<a_{-3}>a_{-2}<a_{-1}>a_0<a_1>a_2<a_3>\cdots,$$right? $\endgroup$ – მამუკა ჯიბლაძე Dec 25 '18 at 18:47
  • $\begingroup$ @მამუკაჯიბლაძე Indeed $\endgroup$ – user49822 Dec 27 '18 at 9:55
  • $\begingroup$ @მამუკაჯიბლაძე what endpoint? $\endgroup$ – user49822 Dec 30 '18 at 17:28
  • $\begingroup$ I was responding to a comment that is no longer there. In any proper connected subspace of your example there are points $k$ such that either $k+1$ or $k-1$ does not belong to the subspace. It is natural (I think) to call these endpoints. If one removes such point from the subspace, what remains will be connected. $\endgroup$ – მამუკა ჯიბლაძე Dec 30 '18 at 17:32
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The other answer describes the "Khalimsky line". It is not $T_1$, but it is possible to obtain Hausdorff examples by starting with a countable connected Hausdorff space $X$, blowing up its points into more copies of $X$, and continuing this process infinitely many times. This ever-branching countable "tree" of $X$'s can be topologized so that it is connected, Hausdorff, and removing any point disconnects the space.

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To be a bit more explicit: in Countable connected spaces Proc. Amer. Math. Soc. 26 (1970) 355-360 G. G. Miller describes a countable connected Urysohn space with a dispersion point.

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    $\begingroup$ That doesn't answer the question - it asks for a space where removal of any point disconnects the space. $\endgroup$ – Wojowu Dec 29 '18 at 20:14
  • $\begingroup$ Indeed, I reacted more to the title than to the content of the question. Still, it indicates that a construction may be possible, though difficult. $\endgroup$ – KP Hart Dec 30 '18 at 22:02
  • $\begingroup$ The answer by @ForeverMozart seems to contain such construction, but I could not see the details, only the devil in them $\endgroup$ – მამუკა ჯიბლაძე Dec 31 '18 at 16:16
  • $\begingroup$ The operative word being `seems' $\endgroup$ – KP Hart Jan 3 at 9:16

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