# Topology Question - Regions

• Dec 14th 2011, 04:54 AM
Speed1991
Topology Question - Regions
Suppose that G is a region and let $\displaystyle a \in G$. Prove that G\{a} is a region.

So far I have divided the proof up into 2 parts, showing that G\{a} is open and showing that it is connected. I have done the open part, but am stuck on the connectedness.
I have assumed for contradiction that G\{a} is disconnected (generally seems to work for this type of question given the definition of connectedness), but cannot see where to get the contradiction from. Help much appreciated.
• Dec 14th 2011, 06:16 AM
Plato
Re: Topology Question - Regions
Quote:

Originally Posted by Speed1991
Suppose that G is a region and let $\displaystyle a \in G$. Prove that G\{a} is a region.

How is region defined?
For example in Moore Spaces regions are points sets.
That is axiom 0.
But in other spaces there are more conditions.
• Dec 17th 2011, 04:46 AM
Speed1991
Re: Topology Question - Regions
A region is defined as a non-empty connected subset of $\displaystyle \mathbb{C}$
• Dec 17th 2011, 07:24 AM
FernandoRevilla
Re: Topology Question - Regions
Quote:

Originally Posted by Speed1991
A region is defined as a non-empty connected subset of $\displaystyle \mathbb{C}$

If $\displaystyle G\subset \mathbb{C}$ is an open set, then $\displaystyle G$ is connected if and only if any two points of $\displaystyle G$ may be connected by a curve in $\displaystyle G$ (in fact we can connect any two points of $\displaystyle G$ by a chain $\displaystyle \Gamma$ of horizontal and vertical segments lying in $\displaystyle G$) . Now, consider the cases $\displaystyle a\not\in \Gamma$ and $\displaystyle a\in\Gamma$ .