In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is said to be a Volterra space if any finite intersection of
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
Gδ subsets is dense. Every
Baire space is Volterra, but the converse is not true. In fact, any
metrizable Volterra space is Baire.
The name refers to a paper of
Vito Volterra in which he uses the fact that (in modern notation) the intersection of two dense G-delta sets in the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
is again dense.
References
* Cao, Jiling and Gauld, D, "Volterra spaces revisited", ''J. Aust. Math. Soc.'' 79 (2005), 61–76.
* Cao, Jiling and Junnila, Heikki, "When is a Volterra space Baire?", ''Topology Appl.'' 154 (2007), 527–532.
* Gauld, D. and Piotrowski, Z., "On Volterra spaces", ''Far East J. Math. Sci.'' 1 (1993), 209–214.
* Gruenhage, G. and Lutzer, D., "Baire and Volterra spaces", ''Proc. Amer. Math. Soc.'' 128 (2000), 3115–3124.
* Volterra, V., "Alcune osservasioni sulle funzioni punteggiate discontinue", ''Giornale di Matematiche'' 19 (1881), 76–86.
Properties of topological spaces
{{topology-stub