Understanding the Relationship between Weak and Strong Topologies

In summary, the problem asks for a connection between "weak" and "strong" topologies, but does not provide a clear explanation of what these terms mean.
  • #1
shinobi20
267
19
Homework Statement
Verify that the “weakest” (coarsest) possible topology on a set ##X## is given by the trivial topology, where ∅ and ##X## represent the only open sets available, whereas the “strongest” (finest) topology is the discrete topology, where every subset is open.
Relevant Equations
1. ∅ ∈ {τ}, ##X## ∈ {τ};
2. the union (of an arbitrary number) of elements from {τ} is again in {τ};
3. the intersection of a finite number of elements from {τ} is again in {τ}.
I do not understand what is to verify here. The problem already defined what it means to be a trivial and discrete topology but it did not state what it means to be "weak" and "strong". I assume the problem wants me to connect "weak" with trivial topology and "strong" with discrete topology, but somehow the problem is not very clear to me or I just do not know how to connect them. Please guide me but do not give me the solution.
 
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  • #2
[itex]T_1[/itex] is weaker than [itex]T_2[/itex] iff [itex]T_1 \subsetneq T_2[/itex] (ie. every set which is open in [itex]T_1[/itex] is open in [itex]T_2[/itex], and there is at least one set which is open in [itex]T_2[/itex] but not in [itex]T_1[/itex].
 
  • #3
pasmith said:
[itex]T_1[/itex] is weaker than [itex]T_2[/itex] iff [itex]T_1 \subsetneq T_2[/itex] (ie. every set which is open in [itex]T_1[/itex] is open in [itex]T_2[/itex], and there is at least one set which is open in [itex]T_2[/itex] but not in [itex]T_1[/itex].
Did you mean ##T_2 \subsetneq T_1##? Clearly (I'll prove this but I'm just clarifying what you said), the trivial topology is a subset of the discrete topology but not the other way.
 
  • #4
pasmith said:
[itex]T_1[/itex] is weaker than [itex]T_2[/itex] iff [itex]T_1 \subsetneq T_2[/itex] (ie. every set which is open in [itex]T_1[/itex] is open in [itex]T_2[/itex], and there is at least one set which is open in [itex]T_2[/itex] but not in [itex]T_1[/itex].

The inclusion need not be strict, or at least this definition is not standard.
 
  • #5
Given a set ##X##, there are multiple possible topologies on ##X##. Say we are given topologies ##\tau_1## and ##\tau_2## on ##X##. Then ##\tau_1 \subseteq \tau_2## means that ##\tau_1## is weaker than ##\tau_2## or equivalently ##\tau_2## is stronger than ##\tau_1##. This is just a definition.

Any set ##X## has a discrete topology ##2^X## (the power set of ##X##, i.e. all subsets of ##X## are open) and an indiscrete topology ##\{\emptyset, X\}##. It is quite trivial to see that these are topologies.

The question asks you to show that if ##\tau## is a topology on ##X##, then ##\{\emptyset, X\}\subseteq \tau \subseteq 2^X## and half a second of thought shows you that this is completely trivial.

I guess your confusion comes from the fact that this "problem" is so trivial that it is not clear what exactly there is to verify.
 
  • #6
shinobi20 said:
Did you mean ##T_2 \subsetneq T_1##? Clearly (I'll prove this but I'm just clarifying what you said), the trivial topology is a subset of the discrete topology but not the other way.

I meant what I said: [itex]\{\emptyset, X\} \subsetneq 2^X[/itex].

Math_QED said:
The inclusion need not be strict, or at least this definition is not standard.

I'm not sure it makes sense to say that a topology is weaker than itself.
 
  • #7
pasmith said:
I meant what I said: [itex]\{\emptyset, X\} \subsetneq 2^X[/itex].
I'm not sure it makes sense to say that a topology is weaker than itself.

Of course that makes sense! Two topologies coincide when one is both weaker and stronger than the other. And take ##X=\{0\}##. The trivial topologies coincide so your strict inclusion is false.
 
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  • #8
pasmith said:
[itex]T_1[/itex] is weaker than [itex]T_2[/itex] iff [itex]T_1 \subsetneq T_2[/itex] (ie. every set which is open in [itex]T_1[/itex] is open in [itex]T_2[/itex], and there is at least one set which is open in [itex]T_2[/itex] but not in [itex]T_1[/itex].
This is contradictory to the statement of @Math_QED, which is

Math_QED said:
Given a set X, there are multiple possible topologies on X. Say we are given topologies τ1 and τ2 on X. Then τ1⊆τ2 means that τ1 is weaker than τ2 or equivalently τ2 is stronger than τ1. This is just a definition.


Math_QED said:
The question asks you to show that if τ is a topology on X, then {∅,X}⊆τ⊆2X and half a second of thought shows you that this is completely trivial.
If this is the question, then a simple proof should go as,

Proof. From the definition that in order to introduce a topology ##τ## on a set ##X##, the sets ##∅## and ##X## must belong to all possible topologies ##τ## of ##X##, then it is guaranteed that ##\{∅, X\}## is a subset of all ##τ## and since ##2^X## contains all subsets of ##X## it is guaranteed that all possible topologies ##τ## is a subset of ##2^X##.

This implies that ##\{∅, X\} ⊆ τ ⊆2^X##. So the trivial topology is the weakest and the discrete topology is the strongest.
 
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  • #9
shinobi20 said:
If this is the question, then a simple proof should go as,

Proof. From the definition that in order to introduce a topology ##τ## on a set ##X##, the sets ##∅## and ##X## must belong to all possible topologies ##τ## of ##X##, then it is guaranteed that ##\{∅, X\}## is a subset of all ##τ## and since ##2^X## contains all subsets of ##X## it is guaranteed that all possible topologies ##τ## is a subset of ##2^X##.

This implies that ##\{∅, X\} ⊆ τ ⊆2^X##. So the trivial topology is the weakest and the discrete topology is the strongest.

Exactly, in other words, if we consider the set of all topologies on ##X## and we partially order it via the inclusion relation, then this set has both a minimum and a maximum.
 
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  • #10
Math_QED said:
Exactly, in other words, if we consider the set of all topologies on ##X## and we partially order it via the inclusion relation, then this set has both a minimum and a minimum.
I understand everything now. Thanks @Math_QED !
 
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Related to Understanding the Relationship between Weak and Strong Topologies

What is the definition of coarsest topology?

The coarsest topology on a set X is the topology that contains only the empty set and the entire set X. In other words, it is the smallest possible topology on X.

What is the definition of finest topology?

The finest topology on a set X is the topology that contains all possible subsets of X. In other words, it is the largest possible topology on X.

How are coarsest and finest topology related?

Coarsest and finest topology are two extremes of the spectrum of possible topologies on a set X. The coarsest topology is the smallest possible topology, while the finest topology is the largest possible topology.

Why are coarsest and finest topology important in topology?

Coarsest and finest topology are important because they serve as reference points for all other topologies on a set. They help us understand the structure of topological spaces and how they relate to each other.

What are some examples of coarsest and finest topology?

An example of coarsest topology is the trivial topology on a set X, which only contains the empty set and the entire set X. An example of finest topology is the discrete topology on a set X, which contains all possible subsets of X.

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