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MATH 730: QUIZ 1
WILLIAM GOLDMAN
Let X be the topological space whose underlying set is R, the set
of real numbers, and whose open sets are the complements of finite
sets. (Equivalently, the closed sets are exactly the finite sets.) Prove
or disprove the following statements:
(1) X is Hausdorff;
(2) X is connected;
(3) X is compact;
(4) X is locally compact;
(5) The identity map R −→ X is continuous;
(6) The identity map X −→ R is continuous;
(7) Every continuous function from X into a metrizable space is
constant.
(8) The addition map
X × X −→ X
(x, y) 7−→ x + y
is continuous.
Date: 19 September 2006.
1
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