E=[: so E is closed in Q. EB(0,2): hence E is bounded in Q. is an open cover of E having no finite subcover. E=( is also open in Q. 21. Construct a compact set of real numbers whose limit points form a countable set. Let E1={1}, E2={1+,…,Em={1+. Consider E=. 22. Give examples to show that for subsets A,B of a MS(X,d), need not imply A=B, A0=B0 need not imply A=B, int (cl A) need not be equal to cl(int A). In Ru, consider subsets Q, R-Q. 23. Let A be a subset of a MS X such that d(A)<r and A. Prove that Let x1AB(x,r). Let yA. Then d(x,y)x1)+d(x1,y)<r+ d(x1,y)r+d(A)<2r. yB(x,2r). 24. Show that for a subset A of a MS X, following are equivalent: (1) A is dense in X, (2) F is closed in X and AF implies F=X, (3)int(X-A)=. Since A is dense in X, every nonempty open set in X intersect A; thus X-A cannot contain any nonempty subset; thus int(X-A)=. (3) Let F be closed in X and AF. Thus X-F is open and contained in X-A. Hence int(X-A) unless X-F=. (2) Let U be open in X and U. Then X-U closed. So X-U=X. Thus U=.Hence A is dense in X. 25. Let aQ and bR-Q, where a,b>0. Show that {xQ:-a<x<a} is open but not closed in the MS Q (with subspace topology)while {xQ:-b<x<b}is clopen in Q.
E=[: so E is closed in Q. EB(0,2): hence E is bounded in Q. is an open cover of E having no finite subcover. E=( is also open in Q. 21. Construct a compact set of real numbers whose limit points form a countable set. Let E1={1}, E2={1+,…,Em={1+. Consider E=. 22. Give examples to show that for subsets A,B of a MS(X,d), need not imply A=B, A0=B0 need not imply A=B, int (cl A) need not be equal to cl(int A). In Ru, consider subsets Q, R-Q. 23. Let A be a subset of a MS X such that d(A)<r and A. Prove that Let x1AB(x,r). Let yA. Then d(x,y)x1)+d(x1,y)<r+ d(x1,y)r+d(A)<2r. yB(x,2r). 24. Show that for a subset A of a MS X, following are equivalent: (1) A is dense in X, (2) F is closed in X and AF implies F=X, (3)int(X-A)=. Since A is dense in X, every nonempty open set in X intersect A; thus X-A cannot contain any nonempty subset; thus int(X-A)=. (3) Let F be closed in X and AF. Thus X-F is open and contained in X-A. Hence int(X-A) unless X-F=. (2) Let U be open in X and U. Then X-U closed. So X-U=X. Thus U=.Hence A is dense in X. 25. Let aQ and bR-Q, where a,b>0. Show that {xQ:-a<x<a} is open but not closed in the MS Q (with subspace topology)while {xQ:-b<x<b}is clopen in Q.
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