Basic questions on Nakahra's definition of topological space

The set {U:U⊆X} is not a topology on X. The definition of a topology requires that the empty set and the whole set are in the topology, but that set doesn't contain the empty set.
  • #1
mjordan2nd
177
1
In chapter 2.3 in Nakahara's book, Geometry, Topology and Physics, the following definition of a topological space is given.

Let [itex]X[/itex] be any set and [itex]T=\{U_i | i \in I\}[/itex] denote a certain collection of subsets of [itex]X[/itex]. The pair [itex](X,T)[/itex] is a topological space if [itex]T[/itex] satisfies the following requirements

1.) [itex]\emptyset, X \in T[/itex]

2.) If [itex]T[/itex] is any (maybe infinite) subcollection of [itex]I[/itex], the family [itex]\{U_j|j \in J \} [/itex] satisfies [itex]\cup_{j \in J} U_j \in T[/itex]

3.) If K is any finite subcollection of [itex]I[/itex], the family [itex]\{U_k|k \in K \} [/itex] satisfies [itex]\cap_{k \in K} U_k \in T[/itex]

I'm not very familiar with mathematical notation and conventions, so this is a little confusing to me. Presently I have a few questions:

1.) What do [itex]I[/itex], [itex]J[/itex], and [itex]K[/itex] denote? I don't think they have been referenced before in the text. Do they just represent integers?

2.) I'm a little confused by the first criterion in the definition. I thought [itex]T[/itex] is a collection of subsets of [itex]X[/itex]. Does the first criterion mean that the set [itex]X[/itex] is in [itex]T[/itex] or that a union of sets in [itex]T[/itex] should yield [itex]X[/itex]?

3.) Why are the last two criterion not considered self evident from the definition of [itex]T[/itex]. For instance, for the second criterion if [itex]J<I[/itex] how can a union of [itex]U_j[/itex] be anything but within [itex]T[/itex]?

Thanks.
 
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  • #2
1) I would assume I, J, and K are sets of integers labeling the open sets in the topological space.

2) T is a collection of subsets of X. It does not mean that T contains all subsets of X. By definition, we call the sets which are in T open sets. The first criterion states that the empty set as well as the full set X are in T.

3) No, these are not self evident. Not by a long shot. It is easy to find collections of sets which this does not hold for. For example, consider the set {1,2,3}. If we would let T contain the set itself, the empty set, {1}, and {2}, it would not satisfy these conditions. In particular, the union of the two last sets would not be in T.
 
  • #3
Ahh, it's not the elements we are concerned about, it's the sets themselves. So for example, {1,2,3} is an element of T, not 1, 2, etc. Thank you, this clarifies things significantly.
 
  • #4
1.) They are simply integers In this case there is a countable collection of subsets. In general the definition of topology is used even if the subset collection is not countable.
2.) The statement simply means that X is a member of T.
3.) They are not self evident. T can be any collection. Simple example. X has three point a,b,c. Let T consist of the following subsets {[itex]\phi[/itex]},{a},{b},{c}, {a,b,c}. This will not be a topology since {a}∪{b}={a,b} is not in T.

To make things clear I am using {...} to mean a subset containing these elements.
 
  • #5
Thank you both for the responses. I feel like I understand now. I was confused about what T was.
 
  • #6
By the way, you should check whether the following Ts satisfy the conditions, it is a good exercise:

1) The set T = {∅,X}.

2) The set T = {U: U ⊆ X}, i.e., T is the set containing all subsets of X.
 
  • #7
I'm not really sure how to write this using proper mathematical notation and terminology so I will primarily use words, but please tell me if this is correct.

It appears to me that both T satisfy the condition.

1.) This satisfies condition 1 since T contains [itex]\emptyset[/itex] and X. This satisfies condition 2 since [itex]\emptyset \cup X = X \in T[/itex]. This satisfies condition 3 since [itex] \emptyset \cap X = \emptyset \in T [/itex].

2.) This satisfies condition 1 since all possible subsets necessarily contain both the empty set and the set itself. This satisfies condition 2 since a union of subsets is always a subset and T contains all possible subset. This satisfies condition 3 since an intersection of subsets is either a subset or the empty set, both of which are contained in T.

Is this correct?
 
  • #8
Yes. Both T are topologies on any space. They are called the trivial and discrete topologies, respectively. When encountering new topological concepts, I found it instructive to check the implications for these two topologies, e.g., checking what it means for a series to converge in a given topology.
 
  • #9
In notations like ##T=\{U_i|i\in I\}##, ##I## is called an index set. It's a set (any set) that can be bijectively mapped onto ##T##. The notation is useful e.g. when the elements of T are subsets of some set X, and you want to say something about a union of some of them. The union of all the elements of T can be denoted by ##\bigcup T##, but the index set gives you the option to write it as ##\bigcup_{i\in I}U_i## instead. If you want a notation for a union of some but not all of them, you can introduce a set ##J## that's a proper subset of ##I##, and write ##\bigcup_{i\in J}U_i##.
 
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Related to Basic questions on Nakahra's definition of topological space

1. What is Nakahra's definition of topological space?

Nakahara's definition of topological space is a set X, together with a collection of subsets of X called open sets, satisfying certain axioms that allow for the study of continuity and convergence of points in X.

2. What are the axioms for a topological space according to Nakahara?

The axioms for a topological space according to Nakahara are: the empty set and the entire space X are open sets, any finite intersection of open sets is also an open set, and any union of open sets is also an open set.

3. How does Nakahara's definition of topological space differ from other definitions?

Nakahara's definition of topological space differs from other definitions in that it places a greater emphasis on the open sets and their properties, rather than the points of the space themselves.

4. What is the significance of Nakahara's definition of topological space in mathematics?

Nakahara's definition of topological space is significant in mathematics because it provides a general framework for the study of continuity and convergence, which are fundamental concepts in many branches of mathematics, including analysis, topology, and geometry.

5. Can Nakahara's definition of topological space be applied to non-mathematical contexts?

Yes, Nakahara's definition of topological space can be applied to non-mathematical contexts, such as in physics, computer science, and social sciences, where the concept of continuity and convergence are relevant. It provides a versatile tool for understanding and analyzing complex systems and structures.

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