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i do not know what this site is expecting to write. i've written my question above . saw in a question paper. again writing it.

is intersection of a countable collection of dense, open subsets of a complete metric space also dense in X?

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Yes, this property is called the Baire property, and every complete metric space is a Baire space

https://en.wikipedia.org/wiki/Baire_category_theorem

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  • $\begingroup$ Yes, i checked it immediately after i got this answer and got my problem solved. thank you very much:) $\endgroup$ – Swati Jul 2 '15 at 5:06

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