Unformatted text preview: er of open sets is open. Give an example to show that the intersection of an infinite number of open sets may not be open. 2. How many distinct topologies are there on a set containing three elements? 3. Show that the interior of a set is open. Show that the closure of a set is closed. Show that A A A. Show that it is possible for A to be empty even when A is not empty. 4. Show that if f : X Y is continuous, and F Y is closed, then f -1(F ) is closed in X.
8 5. Show that a set can be both open and closed. Show that a set can be neither open nor closed. 6. Show that if f : X Y and g : Y Z are both continuous, then g f : X Z is continuous. 7. Show that the two definitions of continuity are equivalent. 8. A subset D X is called dense in X if D = X. Show that it is possible to have a dense subset D with D = . 9. Show that if D is dense in X, then for every open set G X, we have G D = . In particular, every neighborhood of every point in X contains points in D.
9 10. Show that in a Hausdorff space, every set consisting of...
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