Though many safety-critical software systems use floating point to represent real-world input and output, the mathematical specifications of these systems' behaviors use real numbers. Significant deviations from those specifications can cause errors and jeopardize safety. To ensure system safety, some programming systems offer exact real arithmetic, which often enables a program's computation to match its mathematical specification exactly. However, exact real arithmetic complicates decision-making: in these systems, it is impossible to compute (total and deterministic) discrete decisions based on connected spaces such as R. We present programming-language semantics based on constructive topology with variants allowing nondeterminism and/or partiality. Either nondeterminism or partiality suffices to allow computable decision making on connected spaces such as R. We then introduce *pattern matching* on spaces, a language construct for creating programs on spaces, generalizing pattern matching in functional programming, where patterns need not represent decidable predicates and also may overlap or be inexhaustive, giving rise to nondeterminism or partiality, respectively. Nondeterminism and/or partiality also yield formal *logics for constructing approximate decision procedures*. We extended the Marshall language for exact real arithmetic with these constructs and implemented some programs with it.