The University of Waikato Department of Mathematics
Metric Spaces Exercise and Assignment sheet 5:
math501-09B
For the Workshop/Tutorial on 14th August, attempt ques- tions 1, 3, 5(a), 6(a). For assignment 3, due Monday 7th Sept just after the break, do questions 2, 4, 5(b), 6(b).
Metric Topologies:
1. Show that the set of open squares with sides parallel to the axes is a base for the usual topology on R2.
2. Prove that every metric topology is Hausdor.
3. Let X be any set and dene d(x; y) = 1 if x 6= y and d(x; x) = 0 for all x and y. Show that d generates the discrete topology. Show that there is a metric generating the indiscrete topology if and only if the set X has just one point.
4. Prove that if R has the usual topology, every open subset can be written as the countable union of open intervals.
Closure interior and boundary:
5. (a) Prove that @[0; 1) = f0g.
(b) Let R2 have its usual topology generated by open Euclidean balls.
Find the closure, interior and boundary of the set of points f(x; y) : y < 2x 3g:
6. Show that in any topological space and for any subset A, (a) Ao = Ac c and (b) @A = A \ Ac .
14th August 2009
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