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Notes on Topology for Math 501

The basic strategy in real analysis is approximation. In particular, one often tries to approximate “general” elements of some infinite dimensional vector space by “nice” elements from some well understood vector space, possibly even finite dimensional. To quantify approximation schemes involves the consideration of metrics. For example, consider the vector space C([0, 1]) consisting of continuous real valued functions on [0, 1]. The Weirstrass Approximation Theorem says that for any f ∈ C([0, 1]), and any > 0, there is a polynomial p such that