# Connectedness of the complements of the connected subsets

EDIT: My original foolish version was instantly destroyed by Dylan Thurston; it consisted of questions 1 & 2 below. Thus now only new question 0 remains to be answered.

Let $$\ X:=M^n\$$ be a connected manifold (with or without boundary) of dimension $$\ n\ge 1,\$$ or let $$\ X:=S^1\$$ be a circle. Then

$$(E)\qquad X\setminus\{p\}\$$ is connected for every $$\ p\in X$$

Only other hand, of the mentioned spaces only the circle has the stronger property

$$(B)\qquad X\setminus Y\$$ is connected for every connected subspace $$\ Y\subseteq X$$.

Thus $$\ (B)\Rightarrow (E).\$$ Now, given a naturally or classically defined class of topological spaces which already have property $$(E)$$ one would like to prove property $$(B)$$. In particular, let me propose the following two conjectures (and their unspoken about obvious variations; the first conjecture is most likely already know, in which case I would like to ask about references):

0: Let $$\ X\$$ be an arbitrary Hausdorff compact space such that for every open $$\ G\subseteq X\times X,\$$ for which $$\ \forall_{x\in X}(x\ x)\in G,\$$ there exists a continuous surjection $$\ f:X\rightarrow S^1\$$ such that $$\ \forall_{s\in S^1} f^{-1}(s)\times f^{-1}(s)\subseteq G.\$$ Does $$\ X\$$ has property $$(B)\$$ ?   Are such spaces the only ones among Hausdorff connected spaces, which have at least 2 different points, which have property $$(E)\$$ ?

The two questions below were answered (immediately) by Dylan Thurston:

1:   Let $$\ X\$$ be an arbitrary connected topological graph (the body of any finite 1-dimensional simplicial complex) which has no end-points (i.e. $$(E)$$ holds). Does $$\ X\$$ have property $$(B)$$? (answered by Dylan Thurston)

2:   Let $$\ X\$$ be an arbitrary connected 1-dimensional topological space, which has property $$(E)$$. Does $$\ X\$$ have property $$(B)$$? (answered by Dylan Thurston)

About the topological dimension:

Here one may consider one of the three classical topological dimensions: $$\dim$$, ind, or Ind. One may consider separable metric spaces (when the three dimensions are equivalent), or Hausdorf compact spaces for the covering dimension $$\dim$$. Other variations are possible, interesting and welcome.

REMARK 1   For every Hausdorff connected compact space $$\ X,\$$ which has at least two different points, there are at least two different points $$\ p\in X\$$ such that $$\ X\setminus\{p\}\$$ is connected.

REMARK 2   Property $$(B)$$ discussed in this post is antipodal to the notion of the biconnected spaces.

(B) does not hold for topological graphs. Let $X$ be the 1-skeleton of a tetrahedron, and let $S$ be a cycle of 4 edges. Then $X \setminus S$ is not connected.
• In fact, (E) also does not hold for topological graphs without endpoints. Consider the "dumbbell graph" $D$ formed by taking two vertices $x$ and $y$, joining them by an edge, and attaching loops at both $x$ and $y$. (This is a CW complex, and you can subdivide to make this a graph in some more strict sense.) Then $D \setminus \{x\}$ is not connected. – Dylan Thurston May 9 '15 at 17:40