Question

Modified Fort Space Let the set X be the union of any infinite set N and two distinct one point sets {x1} and {x2}. We topologize X by calling any subset of N open and calling any set containing {x1} or {x2} open if and only if it contains all but a finite number of points in N, call this topology T. (a) Is the topological space (X, T) compact? Justify your answer. (b) Is the topological space (X, T) connected? Justify your answer. (c) Is the topological space (X, T) T1? Justify your answer. (d) Is the topological space (X, T) Hausdorff? Justify your answer.

          Modified Fort Space Let the set X be the union of any infinite set N and two distinct one point sets {x1} and {x2}. We topologize X by calling any subset of N open and calling any set containing {x1} or {x2} open if and only if it contains all but a finite number of points in N, call this topology T.

(a) Is the topological space (X, T) compact? Justify your answer.
(b) Is the topological space (X, T) connected? Justify your answer.
(c) Is the topological space (X, T) T1? Justify your answer.
(d) Is the topological space (X, T) Hausdorff? Justify your answer.

Modified Fort Space Let the set X be the union of any infinite set N and two distinct one point sets x1 and x2. We topologize X by calling any subset of N open and calling any set containing x1 or x2 open if and only if it contains all but a finite number of points in N, call this topology T.

(a) Is the topological space (X, T) compact? Justify your answer.
(b) Is the topological space (X, T) connected? Justify your answer.
(c) Is the topological space (X, T) T1? Justify your answer.
(d) Is the topological space (X, T) Hausdorff? Justify your answer.

Added by Elisa B.

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