Limit Points

Prove that the point p is a limit point of the point set X if and only if each open point set containing p contains a point in X which is different from p.

Limit points

Prove that the point p is a limit point of the point set X if and only if each open point set containing p contains a point in X which is different from p. Prove without using sequences. Only use the def. of open set, open interval, and that the point p is a limit point of the point set X means that each open interval containing p contains a point in X which is different from p.

Closure

Prove: If p and q are points, then there exist open point sets U and V containing p and q respectively such that cl(U) and cl(V) are disjoint.

Closure

Prove: If H and K are disjoint closed point sets, then there exist open point sets U and V containing H and K respectively such that cl(U) and cl(V) are disjoint.

Closure

Prove: If H is a closed point set and p is a point of S - H, then there exist open point sets U and V containing H and p respectively such that cl(U) and cl(V) are disjoint.