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Wikibooks has more about this subject:
Topology/Topological Spaces
Wikibooks
NounEdit
standard topology (uncountable)
- (topology) The topology of the real number system generated by a basis which consists of all open balls (in the real number system), which are defined in terms of the one-dimensional Euclidean metric.
- (topology) The topology of a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} such that any subset of that space is open (i.e. belonging to the topology) if it can be written as a union of open balls from that space.
Some "extremal" examples
Take any set X and let = {, X}. Then is a topology called the trivial topology or indiscrete topology.
Let X be any set and let be the set of all subsets of X. The is a topology called the discrete topology. It is the topology associated with the discrete metric.
Remark
A topology with many open sets is called strong; one with few open sets is weak.
The discrete topology is the strongest topology on a set, while the trivial topology is the weakest.
Finite examples
Finite sets can have many topologies on them.
For example, Let X = {a, b} and let ={ , X, {a} }.
Then is a topology called the Sierpinski topology after the Polish mathematician Waclaw Sierpinski (1882 to 1969).
Let X = {1, 2, 3} and = {, {1}, {1, 2}, X}. Then is a topology.
Remark
It is easy to check that the only metric possible on a finite set is the discrete metric. Hence these last two topologies cannot arise from a metric.
The Zariski topology
Let X be any infinite set. Define a topology on X by A if X - A is finite or A = .
This is called the cofinite or Zariski topology after the Belarussian mathematician Oscar Zariski (1899 to 1986)
Examples like this are important in a subject called Algebraic Geometry.
A 'different' topology on R
Let X = R and let = {, R} { (x, ) | x R}
Then is a topology in which, for example, the interval (0, 1) is not an open set.
All the sets which are open in this topology are open in the usual topology. That is, this topology is weaker than the usual topology.
Page 2
We can recover some of the things we did for metric spaces earlier.
Definition
A subset A of a topological space X is called closed if X - A is open in X.
Then closed sets satisfy the following properties
- and X are closed
- A, B closed A B is closed
- {Ai | i I} closed Ai is closed
Proof
Take complements.
So the set of all closed sets is closed [!] under finite unions and arbitrary intersections.
As in the metric space case, we have
Definition
A point x is a limit point of a set A if every open set containing x meets A (in a point x).
Theorem
A set is closed if and only if it contains all its limit points.
Proof
Imitate the metric space proof.
Definitions
The interior int(A) of a set A is the largest open set A,
The closure cl(A) of a set A is the smallest closed set containing A.
It is easy to see that int(A) is the union of all the open sets of X contained in A and cl(A) is the intersection of all the closed sets of X containing A.
Some properties
K1. cl() =
K2. A cl(A) for any subset A
K3. cl(A B) = cl(A) cl(B) for any subsets A and B
K4. cl(cl(A)) = cl(A) for any subset A
Proof
K1. and K2. follow from the definition.
To prove K3. note that cl(A) cl(B) is a closed set which contains A B and so cl(A) cl(A B).
Similarly, cl(B) cl(A B) and so cl(A) cl(B) cl(A B) and the result follows.
To prove K4. we have cl(A) cl(cl(A)) from K2. Also cl(A) is a closed set which contains cl(A) and hence it contains cl(cl(A)).
Remark
These four properties are sometimes called the Kuratowski axioms after the Polish mathematician Kazimierz Kuratowski (1896 to 1980) who used them to define a structure equivalent to what we now call a topology.
Examples
- For R with its usual topology, cl( (a, b) ) = [a, b] and int( [a, b] ) = (a, b).
- In the Sierpinski topology X = {a, b} and ={ , X, {a} } cl({a}) = X and int({b}) =
- If X = R and = {, R} { (x, ) | x R} then cl({0}) = (-, 0] and int( (0, 1) ) =
JOC February 2004
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Metric and Topological Spaces
As previewed earlier whan we considered open sets in a metric space, we can now make the definition:
Definition
A map f: X Y between topological spaces is continuous if f -1(B) X whenever B Y.
Remark
Note that a continuous map f: X Y "induces" a map from Y to X by B f -1(B).
Definition
A map f: X Y between topological spaces is a homeomorphism or topological isomorphism if f is a continuous bijection whose inverse map f-1 is also continuous.
Remark
By the remark above, such a homeomorphism induces a one-one correspondence between X and Y.
Examples
- Let f be the identity map from (R2, d2) to (R2, d). Then f is a homeomorphism.
Proof
Since every open set is a union of open neighbourhoods, it is enough to prove that the inverse image of an -neighbourhood is open. This -neighbourhood is an open square in R2 which is open in the usual metric.
A similar proof shows that the image of an -neighbourhood in the usual metric (an open disc) is open in d .
- In general, if X is a set with two topologies 1 and 2 then the identity map (X, 1) (X, 2) is continuous if 1 is stronger (contains more open sets) than 2 .
JOC February 2004
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