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Let X be an arbitrary topological space. Pick out the true statement(s):

(a) If X is compact, then every sequence in X has a convergent subsequence.

(b) If every sequence in X has a convergent subsequence, then X is compact.

(c) X is compact if, and only if, every sequence in X has a convergent

subsequence.

from point of view my point all option (a) , (b) and (c) are correct. Because in compact subspace all sequence are convergent ,, If anbody help me i would be very thankful to him....

jasmine
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1 Answers1

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Compactness implies sequential compactness for first countable spaces, but not in general. Option a) is therefore incorrect. Conversely sequential compactness implies compactness for second countable spaces, but not in general. Option b) is therefore incorrect. This means that option c) is also incorrect.

Note then that a) is true for first countable spaces and b) and c) are true for second countable spaces.

Tyrone
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  • i think for option (a) cofinite topology i satisfied and for option (b) lower topology is satisfied ,,,,,,is my answer is correct or not @ tyrone – jasmine Aug 17 '17 at 12:50
  • special thanks to u@ tyrone,,,,but pliz give my answer – jasmine Aug 17 '17 at 12:56