When you cut something with a knife, does the blade come into contact with each of the little pieces? I think so.
Now consider a topological version of this question. Let 
be a connected topological space (with more than 1 point) which we “cut” by removing a point 
. Let 
be a connected component of 
; this is one of the little pieces. Is it true that 
?
True if we replace “connected” by “path-connected”. Indeed, let ![\gamma\colon [0,1]\to X](http://s0.wp.com/latex.php?latex=%5Cgamma%5Ccolon+%5B0%2C1%5D%5Cto+X&bg=ffffff&fg=333333&s=0 "\gamma\colon [0,1]\to X")
be a continuous function such that 
and 
. If necessary, we truncate 
so that 
for 
. Now the set 
is a path-connected subset of 
, hence 
. Since 
as 
, we have .
False as stated, even if is a subset of 
. Here is a counterexample given by Niels Diepeveen on Math.StackExchange:
Counterexample by Niels Diepeveen
The set consists of the point 
and of closed line segments from 
to 
. The union of line segments is connected, and since it is dense in , is also connected. However, once 
is removed, each line segment becomes its own connected component, and so does the point . Clearly, the closure of 
does not contain .
In this example is not compact. It turns out that the statement is true when is a compact connected Hausdorff space (i.e., a continuum), but the proof (also given by Niels Diepeveen) is not easy.
Have you heard of the Knaster-Kuratowski fan? It is a connected subset of the plane that becomes totally disconnected upon removal of a certain point. So the knife doesn’t touch any of the pieces that remain.
That’s a more impressive example, of course — but clearly beyond my ability to draw…