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Is the set of all polynomial open in the space $C[-1, 1]$?

2 Answers 2

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No. Any open set that contains $0$ contains a ball around $0$. The polynomials are a subspace, so if they contain a ball around $0$ they have to be the whole space.

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No, in a ball around any polynomial you can find a continuous function that is not a polynomial. For instance by adding a suitably small bump function to it.