F-Spaces and Substonean Spaces General Topology as a Tool in Functional Analysis

K. Grove and G. Pedersen define a substonean space to be a locally compact (Hausdorff) space in which disjoint σ-compact open subspaces have disjoint compact closures. It is routine to verity that a locally compact space X is substonean if and only if every continous f: S → K, where K is a compact, has a unique continuous extension f: Cℓ_xS → K whenever S is a σ-compact open subspace of X. Spaces with the property obtained by deleting σ-compact in the above are called stonean spaces and must b compact. If the only requirement is that open subspaces have open closures, such spaces are said to be extremally disconnected. Thus, a spacie is stonean if and only if is compact and extremally disconnected