Connected (topology)
Definition
A topological space [math](X,\mathcal{J})[/math] is connected if there is no separation of [math]X[/math]
Separation
This belongs on this page because a separation is only useful in this definition.
A separation of [math]X[/math] is a pair of two non-empty open sets [math]U,V[/math] where [math]U\cap V=\emptyset[/math] where [math]U\cup V=X[/math]
Equivalent definition
We can also say: A topological space [math](X,\mathcal{J})[/math] is connected if and only if the sets [math]X,\emptyset[/math] are the only two sets that are both open and closed.
Theorem: A topological space [math](X,\mathcal{J})[/math] is connected if and only if the sets [math]X,\emptyset[/math] are the only two sets that are both open and closed.
Connected[math]\implies[/math]only sets both open and closed are [math]X,\emptyset[/math]
- Suppose [math]X[/math] is connected and there exists a set [math]A[/math] that is not empty and not all of [math]X[/math] which is both open and closed. Then as :this is closed, [math]X-A[/math] is open. Thus [math]A,X-A[/math] is a separation, contradicting that [math]X[/math] is connected.
Only sets both open and closed are [math]X,\emptyset\implies[/math]connected
TODO:
Connected subset
Let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] be two topological subspaces - they are separated if each is disjoint from the closure of the other (closure in [ilmath]X[/ilmath]), that is:
- [math](B\cap \bar{A})\cup(A\cap\bar{B})=\emptyset[/math]
