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.

Re: Topology Question - Regions

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**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.

Re: Topology Question - Regions

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

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 $\displaystyle a\in\Gamma$ .