Basic questions on Nakahra's definition of topological space

[mjordan2nd]

In chapter 2.3 in Nakahara's book, Geometry, Topology and Physics, the following definition of a topological space is given.

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

1.) $\emptyset, X \in T$

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

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

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 $I$, $J$, and $K$ 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 $T$ is a collection of subsets of $X$. Does the first criterion mean that the set $X$ is in $T$ or that a union of sets in $T$ should yield $X$?

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

Thanks.

 

[Orodruin]

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.

 

[mjordan2nd]

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.

 

[mathman]

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 {$\phi$},{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.

 

[mjordan2nd]

Thank you both for the responses. I feel like I understand now. I was confused about what T was.

 

[Orodruin]

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.

 

[mjordan2nd]

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 $\emptyset$ and X. This satisfies condition 2 since $\emptyset \cup X = X \in T$. This satisfies condition 3 since $\emptyset \cap X = \emptyset \in T$.

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?

 

[Orodruin]

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.

 

[Fredrik]

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 $\bigcup_{i\in J}U_i$.