The forgetful functor Γ:Top→Set\Gamma : Top \to Set from Top to Set that sends any topological space to its underlying set has a left adjoint Disc:Set→TopDisc : Set \to Top and a right adjoint Codisc:Set→TopCodisc : Set \to Top.
[Disc⊣Γ⊣Codisc]:Top←Codisc→Γ←DiscSet. [Disc \dashv \Gamma \dashv Codisc] : Top \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \,.
For S∈SetS \in Set
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Disc[S]Disc[S] is the topological space on SS in which every subset is an open set,
this is called the discrete topology on SS, it is the finest topology on SS; Disc[S]Disc[S] is called a discrete space;
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Codisc[S]Codisc[S] is the topological space on SS whose only open sets are the empty set and SS itself, which is called the indiscrete topology on SS [rarely also antidiscrete topology or codiscrete topology or trivial topology or chaotic topology [SGA4-1, 1.1.4]], it is the coarsest topology on SS; Codisc[S]Codisc[S] is called a indiscrete space [rarely also antidiscrete space, even more rarely codiscrete space].
For an axiomatization of this situation see codiscrete object.
Properties
The left adjoint of the discrete space functor
The functor DiscDisc does not preserve infinite products because the infinite product topological space of discrete spaces may be nondiscrete. Thus, DiscDisc does not have a left adjoint functor.
However, if we restrict the codomain of DiscDisc to locally connected spaces, then the left adjoint functor of DiscDisc does exist and it computes the set of connected components of a given locally connected space, i.e., is the π 0\pi_0 functor.
This is discussed at locally connected spaces – cohesion over sets and cosheaf of connected components.
References
For Grothendieck topologies, the terminology “chaotic” is due to
reviewed, e.g., in:
- The Stacks Project, Example 7.6.6
Conceptualization of the terminology via right adjoints to forgetful functors [see also at chaos] is due to
- William Lawvere, Functorial remarks on the general concept of chaos IMA preprint #87, 1984 [pdf]
and via footnote 1 [page 3] in:
- William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, preprint, State University of New York at Buffalo, [1986] Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 [tac:tr9, pdf].
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.
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.
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.
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
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
Page 3
Metric and Topological SpacesAs 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