Let A be a connected subset of R^n. Let p be a limit point of A. Show that A U {p} is connected.
Helping with proofs involving connected sets is very difficult without knowing the exact definition and sequence of theorems in your textbook. There are many different but equivalent approaches to connected sets. This particular question is either a simple consequence of the theorem- “The closure of a connected set is connected”. Or it may be a lemma to the proof of that theorem. Basically you would show that if Au{p} is separated some open set contains p and no other point of A. That is a contradiction.
Well, I'm trying to solve it this way:
A is a connected set, implies A has Intermediate value property, so for each continuous function f:A->A, f(A) is an interval.
Now p is a limit point of A, so there exist a sequence {Pn} in A\{p} such that {Pn} -> p.
And I'm thinking if I can suppose to the contrary that A is not connected, then the function from {Pn} to p cannot be an interval, in which is a contradiction.
But something doesn't look right. How do I know that Pn to p is continuous?
Or should I use the seperate operators? Say there are U and V in R^n that seperates AU{p}, then I can try to get a contradiction. Oh yeah, that sounds a lot more convincing.
I will say again: Helping with proofs involving connected sets is very difficult without knowing the exact definition and sequence of theorems in your textbook. There are many different but equivalent approaches to connected sets.
You may be using a completely non-standard set of definitions. That happens: in Moore spaces compact set are not necessarily closed.
But it seems to me that you should at least know the definitions: The closure of a set is the set union all its limit points. So if p is a limit point of A then p is in A closure. If every open set about p contains a point of A distinct from p, then how could p possibly be separated from A? Just look at the definition for connectivity.