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Chapter 2 TYPES OF TOPOLOGY Topology has been established as one of the basic disciplines of pure mathematics

May 1, 2019 0 Comment

Chapter 2
TYPES OF TOPOLOGY

Topology has been established as one of the basic disciplines of pure
mathematics .It has greatly enlarged the growth of abstract algebra. There are
many things that in the exact field of topology. All the topological concepts are a
base to mathematics of today as the sets and functions were to those of last century
.It has more branches like general topologywhich is the door to the study of
others,algebraic topology, differential topology…etc. Here I aimed in this chapter
to provide a through grounding in the ty pes of topology discrete and usual
topology.

1. Discrete Topology

In topology, a discrete space is a particularly simple example of a topological
space. The discrete topology is the finest topology that can be given on a set, i.e. it
defines all subsets as open sets. In particular, each singleton is an open set in the
discrete topology.

Example:

Given X= { 1 , 2 , 3 , 4 } and the collection T=P(X) The
power set of X, (X, T) is a topological space. T is called the discrete
topology.

Discrete topology, discrete uniformity and discrete metric

The discrete topology on X is defined by letting every subset of X be
open (and hence also closed), and X is a discrete topological space if it is equipped
with its discrete topology.

The discrete uniformity on X is defined by letting every superset of
the diagonal { ( x , x ) : x is in X} in X×X been an entourage, and is a
discrete uniform space if it is equipped with its discrete uniformity.

The discrete metric µ on X is defined by

µ(x, y) = { 1 if x ? y,
0 if x=y

Fur any x, y in X. In this case (X, µ) is called a discrete metric space or a space of
isolated points.
A set S is discrete in a metric space (X, d), for S is the subset o f X, if for every x
in S, there exists some ?>0 such that d(x, y) >? for all y in S {x};such a set
consists of isolated points .A set S is uniformly discrete in the metric space (X,

d), for S in X ,if there exists ?>0 such that for any two distinct x, y be longs to S,
d{x, y)>?.

USES

A discrete structure often used as the “default structure” on a set that does not
carry any other natural topology, uniformity, or metric; discrete structures can
often be used as “extreme” examples to test particular suppositions. For example,
any group can be considered as a topological group by giving it the discrete
topology, implying that theorems about topological groups apply to all groups.
In deed, analysts may refer to the ordinary, non -topological groups studied by
algebraists as “discrete groups”. In some cases, this can be usefully appl ied.

A product of infinite copies of the discrete space of natural numbers is
homeomor phism to the space of irrational numbers, with the homeomorphism
given by the continued fraction expansion. A product of infinite copies of the
discrete space {0, 1} is homeomorphism to the cantor set; and in fact uniformly
homeomorphism to the cantor set if we use the product uniformity on the product.
Such a homeomorphism is given by using ternary notation of numbers.

In the foundation of mathematics, the study of compactness properties of products
of {0, 1} is central to the topological approach to th e ultra filter principle, which is
a weak form of choice.

Additionally:

1. The topological dimension of a discrete space is equal to 0.

2. A topological space is discrete if and only if its singletons
ar e opens, which is the case if and only if it does not contain
any accumulation points.

3. The singletons form a basis for the discrete topology.

4. The uniform space X is discrete if and only if the diagonal
{(x, x): x is in X} is an entourage.

5. A discrete space is compact if and only if it is finite.

6. Every discrete uniform or metric space is complete.

7. Every discrete metric space is bounded.

8. Every discrete space is totally disconnected.

9. Any two discrete spaces with the same cardinality are
homeomorphis m.

2. Usua l Topology

If we make all possible combinations of all open intervals in R and the
union on each combination, we get the usual topology on R. Open interval is a
concept we all learned at high school, which I think is intuitive enough.

It is important because it is probably the most usual topological space in
the real world. Anything whose element can be abstracted as open interval or union
of open intervals is an instance of the topology on R.

The usual topology on R is actually def ined as the topology induced, by the
Euclidean metric .We know that, with this metric the open balls are just bounded
open intervals. So all bounded open intervals can be expressed as unions of
bounded open intervals, they are also open in the usual to pology. Thus all open
intervals are open sets of R in the usual topology. But the converse is not true.

* What is the difference between the discrete topology and usual topology?

” We can define a topology T on R by defining a subset U of R to be in T if for
every point x in U, there is a ?>0 such that (x -?, x+ ?) is a subset of u. We call this
topology the standard topology or usual topology on R. ”

In the discrete topology EVERY subset of R is open. In the
usual topolog y only open intervals and unions of open
intervals are open. And sets like 0, 1, Q, {2}, etc. Are not
open in the usual topology, but are open in the discrete
topology….