In the above dictionary definition distance is defined as the extent of spatial
separation between objects. Well, the amount of separation can be anything from infinitely small
III. Convergence of series and sequences of functions.
to infinitely large. That implies a continuum of possible distances. If a distance concept exists
that is similar to the familiar distance from three dimensional space, then a continuum of
distances will exist. If a distance concept doesn’t exist, a continuum concept can’t exist.
In order for a sequence to converge in a complete metric space it is necessary and sufficient that it be a Cauchy sequence. The idea was then conceived of defining a “distance” with these algebraic properties in spaces
like n-dimensional space that don’t possess a natural concept of distance. Thus n-space in which
a “distance” has been defined was presumably the first “metric space” (a space other than regular
space containing an artificially defined distance). It was then realized that one could do the same
thing with other spaces (such as functional spaces) and the mathematical structure of
axiomatically defined metric space was conceived.
The theory of statistical convergence has many applications in various fields such as approximation theory , finitely additive set functions , trigonometric series , and locally convex spaces . Mathematicians in ancient times (Euclid, Archimedes) used the concept of convergence in using series to find areas and volumes. They reasoned that by the method of exhaustion, they could prove the convergence of series. The term “convergence” was introduced in the context of series in 1668 by J. Gregory in his research on the methods of calculating the area of a disc and of a hyperbolic sector.
Consequently we have generalised several results in cone metric spaces from metric spaces. Every metric space is a cone metric space, and every cone metric space is a topological space. In this paper, we introduce and investigate statistical convergence in cone metric spaces, discuss statistically-sequentially compact spaces and characterize statistical completeness of cone metric spaces. Having defined convergence of sequences, we now hurry on to define continuity for functions as well.
Ε-neighborhood of a point. The ε-neighborhood of a point P is the open set
Relation between multivariate and univariate convergence
consisting of all points whose distance from P is less than ε. We note that the boundary is not
included. Another name for ε-neighborhood is open sphere.
It is natural to wonder if we could interpret them as a four dimensional continuum similar to the
three dimensional continuum of 3-space. However, in the case of the points in 3-space there is a
natural distance defined between points but in the case of the set of the points (x1, x2, x3, x4) this
is not so. Without a distance how can they form a continuum?
Now the intersection
of two disjoint closed sets is also the null set ∅. Thus, by Theorem 7, the null set ∅ is closed. From this we deduce from Theorem convergence metric 5 that the full set M is open. A closed sphere is a sphere that contains all of its limit points i.e. it is an open
sphere plus its boundary points.
- The following definition formalizes what we have just said.
- The equivalence between these two definitions can be seen as a particular case of the Monge-Kantorovich duality.
- For insight and clearer understanding of the concepts
look to the point sets of one, two and three
dimensional spaces for a model from which to think.
- Then, in 2014, Zhou et al.  generalized the notion of PM-space to the G-metric spaces and defined the probabilistic generalized metric space which is denoted by PGM-space.
- Let M be the real line and τ be the set of all open sets
Similarly we also define convergence. Again, we will be cheating a little bit and we will use the definite article in front of the word limit before we prove that the limit is unique. In other words, a function f is continuous if and only if the inverse of each open set in the range
R is open in the domain D (or if and only if the inverse of each closed set in R is closed in D). The subject of topology is much interested with continuous
mappings of figures. 14 a sphere is carried into a pear shape by a bicontinuous, bijective
(one-to-one, onto) mapping.
Cauchy’s condition for convergence. Cantor’s Intersection Theorem. A limit of a sequence (or net) of points (x i)(x_i) in a topological space (or other convergence space) XX is a point xx such that the sequence eventually gets arbitrarily close to xx. We can also speak of a limit of a filter on XX. The equivalence between these two definitions can be seen as a particular case of the Monge-Kantorovich duality.
Otherwise, the sequence is said to be divergent. In regular continuity δ depends not only on ε but also on the point p0. Uniform continuity is
essentially continuity plus the added condition that for each ε we can find a δ which works
uniformly over the entire space X, in the sense that it does not depend on p0. A set E in a space M is
nondense (or nowhere dense) relative to M if the
closure of E contains no interior points, or
(equivalently) if the complement of the closure of E is dense in M.