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 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 ...continues
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.
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.
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.
If M is a nondegenerate connected point set, then is each point in M a limit point of M?
Prove true or false, if false give a counter example. If each point in the point set M is a limit point of M, then M is connected.
Understanding connected point sets.
Prove whether the following is true or false. If it is false give a counter example. If M is a connected point set, the cl(M) is connected.
Understanding connected point sets.
Prove whether the following is true or false. If it is false give a counter example. If F is a collection, each element of which is a connected point set and each point set in F contains a limit point of every other point set in F, then UF is connected.
Understanding connected point sets.
Prove whether the following is true or false. If it is false give a counter example. If F is a collection, each element of which is a point set such that if X and Y are elements of F then X U Y is connected, then UF is connected.
