Let κ be an uncountable cardinal with κ~(＜κ) = κ. In this paper we introduce R_κ, a Cauchy-complete real closed field of cardinality 2~κ. We will prove that R_κ shares many features with R which have a key role in real analysis and computable analysis. In particular, we will prove that the Intermediate Value Theorem holds for a non-trivial subclass of continuous functions over R_κ. We propose R_κ as a candidate for extending computable analysis to generalised Baire spaces.
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