Descriptive complexity of function spaces

Lutzer, D.; van Mill, J.; Pol, R.

doi:10.1090/S0002-9947-1985-0797049-2

Descriptive complexity of function spaces
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by D. Lutzer, J. van Mill and R. Pol PDF

Trans. Amer. Math. Soc. 291 (1985), 121-128 Request permission

Abstract:

In this paper we show that ${C_\pi }(X)$, the set of continuous, real-valued functions on $X$ topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in ${{\mathbf {R}}^X}$ even when $X$ is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces $X$ for which ${C_\pi }(X)$ lies nowhere in the projective hierarchy of the complete separable metric space ${{\mathbf {R}}^X}$.

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Measure and category