4. Consider the following axiom system: Undefined Terms: point and region. Axiom

Question

4. Consider the following axiom system: Undefined Terms: point and region. Axiom: Every region is a set consisting of points. Definitions: A point p is a limit point of a region R if and only if every region containing p contains points of R other than p. a. b. A point p is a boundary point of a region R if and only if every region containin p contains points of R other than p and points not in R. C. Based on these axioms and definitions, prove the theorems below. xsmEvery bounda point of regin Ris a imni oint o Theorem 2: No point p of a region R is a boundary point of R.

Explanation / Answer

Theorem 1:Every boundary point of a region is a limit point of R

Proof:

Let P be a boundary point of the region. Then the region surrounding P contains points of R other than P and also points not in P.

In other words, the point P has a neighbourhood, which contains both points of R and also other than R

In which neighbourhood, the point P contains other than R, definitely is boundary for R.

Since every neighbourhood of P contains points other than R and also of R, we have the point P has to lie in the boundary, because if not, then between P and the boundary we can find a neighbourhood of P, which do not contain points other than R, which leads to a contradiction.

Hence proved

Theorem 2:

No point p of a region is a boundary point of R

Let p be a point of the region.

This implies p contains a neighbourhood which contains only points of R

This implies that p is in the interior of the region R.

Hence p cannot have all neighbourhoods having points other than that of R

So p cannot be a boundary point.

Thus proved

i.e.

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