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In Kreyszig compactness of metric space X is defined as :

A metric space X is said to be compact if every sequence in X has a convergent subsequence.

My confusion is simply because of the use of the if in the statement. Consider the sentence:

Statement A if Statement B.

This means that if Statement B is true then Statement A is true, but if Statement B is false, Statement A may or may not be true (see here). I feel that the if in the definition should be replaced by the only if.

amitoz
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    I think it is common to simply write “if” for definitions. – nejimban Jan 24 '22 at 16:37
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    Here your first statement "$X$ is compact" would not be an actual statement, since the term "compact" is not defined yet. As nejimban said, it is an usual way to write definitions. – Balloon Jan 24 '22 at 16:40

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