Page 54 - DMTH401_REAL ANALYSIS

Real Analysis

Notes where the infimum is taken over all finite sequences (p , p , . . ., p ) and (q , q , . . ., q ) with [p ]
1 2 n 1 2 n 1
= [x], [q ] = [y], [q ] = [p + 1], i = 1, 2, . . ., n – 1. In general this will only define a pseudometric, i.e.
n i i
d’([x], [y]) = 0 does not necessarily imply that [x] = [y]. However for nice equivalence relations
(e.g., those given by gluing together polyhedra along faces), it is a metric. Moreover if M is a
compact space, then the induced topology on M/~ is the quotient topology.
The quotient metric d is characterized by the following universal property. If f : (M, d)  (X, )
is a metric map between metric spaces (that is, (f(x), f(y))  d(x, y) for all (x, y) satisfying
( f x
f(x) = f(y) whenever x  y, then the induced function f : M/ X, given by [ ]) = f(x) , is a
metric map f : (M/ , d')  (X, ) . A topological space is sequential if and only if it is a quotient

of a metric space

Generalizations of Metric Spaces

 Every metric space is a uniform space in a natural manner, and every uniform space is
naturally a topological space. Uniform and topological spaces can therefore be regarded
as generalizations of metric spaces.
 If we consider the first definition of a metric space given above and relax the second
requirement, or remove the third or fourth, we arrive at the concepts of a pseudometric
space, a quasimetric space, or a semi-metric space.
 If the distance function takes values in the extended real number line R{+}, but otherwise
satisfies all four conditions, then it is called an extended metric and the corresponding
space is called an -metric space.
 Approach spaces are a generalization of metric spaces, based on point-to-set distances,
instead of point-to-point distances.
 A continuity space is a generalization of metric spaces and posets, that can be used to unify
the notions of metric spaces and domains.

Metric Spaces as Enriched Categories

The ordered set (, ) can be seen as a category by requesting exactly one morphism a  b if
a  b and none otherwise. By using + as the tensor product and 0 as the identity, it becomes a
monoidal category R*. Every metric space (M, d) can now be viewed as a category M* enriched
over R*:
 Set Ob(M*): M=

 For each set X, Y  M set Hom(X, Y) : = d(X, Y) Ob(R*).
 The composition morphism Hom(Y, Z)  Hom(X, Y)  Hom(X, Z) will be the unique
morphism in R* given from the triangle inequality d(y, z) + d(x, y)  d(x, z).

 The identity morphism 0  Hom(X, X) will be the unique morphism given from the fact
that 0  d(X, X).
 Since R* is a strict monoidal category, all diagrams that are required for an enriched
category commute automatically.
3.2 Modulus of Real Number

You know that a real number x is said to be positive if x is greater than 0. Equivalently, if 0
represents a unique point 0 on the real line, then a positive real number x lies on the right side
of 0. Accordingly, we defined the inequality x > y (in terms of this positivity of real numbers) if

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