Webbed space
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a "web" that satisfies certain properties. Webs were first investigated by de Wilde. Web. Let "X" be a Hausdorff locally convex topological vector space. A web is a stratified collection of disks satisfying the following absorbency and convergence requirements . The first stratum must consist of a sequence of disks in "X", denoted by formula_1 such that formula_2. For each disk formula_3 in the first stratum, there must exists a sequence of disks in "X", denote by formula_4 such that and formula_7 absorbs formula_8 This sequence of sequences will form the second stratum. To each disk in the second stratum another sequence of disks with analogously defined properties can be assigned. This process continuous for countably many strata. A strand is a sequence of disks, with the first disk being selected from the first stratum, say formula_3, and the second  being selected from the sequence that was associated with formula_3, and so on. We also require that if a sequence of vectors formula_11 is selected from a strand (with formula_12 belonging to the first disk in the strand, formula_13 belonging to the second, and so on) then the series formula_14 converges. A Hausdorff locally convex topological vector space on which a web can be defined is called a webbed space. Examples and sufficient conditions. All of the following spaces are webbed: Theorems. If the spaces are not locally convex, then there is a notion of web where the requirement of  being a disk is replaced by the requirement of being balanced. For such a notion of web we have the following results: