Number of points that allow a topological space to stay connected This question stems from a problem a friend of mine in the software field posed with regards to a graph. I am curious as to whether there is some analogue for topological spaces in general , maybe with some restrictions imposed.
He defined a $\textbf{critical node}$ as a vertex which allows a given graph to stay as one connected piece. He also defined a $\textbf{critical set}$ as a set of vertices which together ensure the same thing that a critical node does. 
The question is this:
Given a graph is there a closed expression that allows one to speculate on the number of such critical nodes and critical sets. Maybe a bound??
Further I was wondering whether something similar exists for topological spaces in general.
I am not familiar with much algebraic topology except for knowing the fact that path-connectedness is homotopy invariant.
I hope this is a sensible question.Thank you for the help.
 A: This is a collection of thoughts on your question, rather than a real answer. The only real answer I have to the question about graph theory is that there's probably a great deal of research already done to determining things like that, since graph theory is important for computer science. For instance, many of the notions on this Wikipedia page are related to your question, and the theorems and definitions there may give you what you want.
As far as generalizing these ideas to other spaces, I agree that things like the real line are probably not the best objects to look at, but here's something you could potentially think about. Graphs are actually a special case of a topological object called a simplicial complex. They're built from points and edges, like graphs, but can also have higher dimensional cells. 
Now all spaces have a collection of things called homotopy groups that measure how "connected" the space is. The level-zero group, called $\pi_0(X)$, measures how many path components the space has, and isn't really a group at all, but that's not important. It seems like what you could do is take your simplicial complex, and to each cell $C$ look at the inclusion map $i:X\setminus st(C) \rightarrow X$ from the space without that cell's star into the big space. This gives a map of the sets $\pi_0(X\setminus st(C)) \rightarrow \pi_0(X)$. If that cell was important for holding the space together in one piece, then this map shouldn't be an isomorphism, since that cell is "gluing" together several pieces of the old graph. So you might call that cell "critical". Perhaps call it "0-critical", if you want to sound fancier. You could also take stars of whole sets. So if you looked at these maps for various collections of simplices, maybe that would be something along the lines of what you're after.
There are more groups, $\pi_1(X), \pi_2(X)$, and so on, but they get complicated. $\pi_1(X)$ basically measures how many "1-dimensional holes" the complex has. For instance, if you built a circle out of these, then $\pi_1$ would be $\mathbb{Z}$. The torus has $\pi_1(torus) = \mathbb{Z}^2$, because there are two "loops". Hope this is helpful in pointing you in the right direction!
