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.
