Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971

FUNCTIONS ON Κ-Net CONVERGENCE STRUCTURES

“…E. Hodel also introduces a convergence structure on a set X that is defined in terms of κ-nets and then gives a cryptomorphic description of κ-net spaces and κ-Fréchet spaces in terms of certain axioms for these convergence structures. These results extend the classical study ( [2,4,5]) of sequential and Fréchet spaces in terms of sequential convergence structures to higher cardinality.…”

On Uniform Lipschitz-Connectedness in Metric Spaces

“…A subset C of X is then said to be Lipschitz-connected if any two points in C can be joined by a Lipschitz path that lies entirely in C. A subset C of a metric space (X, d ) is uniformly Lipschitz-connected if there exists a positive constant L such that for any two points x 0 and x 1 in C we can find a path g : [0, 1] −→ C with g(0) = x 0 and g(1) = x 1 such that d(g(s), g(t)) ≤ L|s − t|d(x 0 , x 1 ) for all s, t ∈ [0, 1] [1]. For the basic categorical concepts we consider in this paper we refer the reader to [2].…”

A Note on Locally Pathwise Connected Metric Spaces

“…A subset C of X is pathwise connected if any two points in C can be be joined by a path that lies entirely in C. By d(C) we mean the diameter of the subset C relative to the metric d. The open sphere centred at x radius r in a metric space (X, d) is denoted by S d (x, r), or S(x, r) if the context is clear which metric is being considered. For basic categorical concepts we refer to [1], and for basic topological concepts we refer to [3]. done.…”