How Can a Finite Set of Natural Numbers Be an Open Set in Topology?

[0ddbio]

So I want to start learning about topological spaces, however I couldn't even get past the definition.

My book states:

DEFINITION (Open sets, neighborhoods) Let E be an arbitrary set. A topology on E is the data of a set O of subsets of E, called the open subsets of E (for the given topology), which satisfy the following three properties:
i) $$\emptyset \in O$$ and $$E \in O$$

I won't write the other two, but the thing I do not understand is how on Earth can O contain open sets?

I checked wikipedia and they give basically the same definition as my book, but they have examples, one example is that

X={1, 2, 3, 4} with the collection τ={{}, {1, 2, 3, 4}} form a topology.
So I guess τ is like the O in my books definition... but again, how is it possible that {1, 2, 3, 4} is an open set? I thought that an open set requires there to be an $\epsilon>0$ that can be added to any element in the open set such that the result is still in that open set.

However... I can think of many epsilons that can be added to elements in {1, 2, 3, 4} that do not give elements of the set, so why is it considered an open set?Basically my question is... how is it possible for a finite set of natural numbers to be an open set?

 

[CompuChip]

O contains the open sets by definition.
If you have a topological space X, then the definition says: of all the possible subsets of X, there are some that we assign a special property, which is that we call them "open".

Your confusion arises because of the fact that you are used to working with spaces like R, which have another definition of when we call any arbitrary subset of it "open".
However, as you say, for this definition you need to take any $\epsilon > 0$ and then you need to speak about the distance $d(x, y) = |x - y|$ between real numbers.

The idea is that in a general topological space, such a distance need not be defined. For example, I could define X = { apple, pear, grape, orange } and say: the open sets of X are defined to be the empty set {} and { apple, pear, grape, orange }. However, I cannot use the epsilon definition here because I have not defined a distance function (the statement $|\text{apple} - \text{orange}| < \epsilon$ is meaningless).

If you just bite the bullet and accept the definition, you will be shown later that in special spaces where a distance function is defined (such as R) the toplogical and analytical definitions agree on what sets are called open.

 

[Greg Bernhardt]

First of all, don't get discouraged! Topology can be a difficult subject to understand at first, but with some patience and practice, it will start to make more sense.

To answer your question, it might help to think of open sets in a more abstract way. In topology, open sets are not necessarily defined by an epsilon value or any sort of distance metric. Instead, open sets are defined by their properties and how they interact with other sets in the topology.

In the example you mentioned, the set {1, 2, 3, 4} is considered an open set because it satisfies the properties of an open set in the given topology. In this case, the only two open sets are the empty set and the entire set {1, 2, 3, 4}. This might seem strange, but remember that topologies can vary greatly and can have different sets of open sets depending on how they are defined.

Think of it this way: in the real number line, the open interval (0, 1) is considered an open set. But in the discrete topology on the real line, where every subset is open, the set {0, 1} is also considered an open set. The key is that these sets satisfy the properties of open sets in their respective topologies.

So, in summary, don't get too hung up on the idea of an open set needing to have an epsilon value or a specific distance metric. Instead, focus on understanding the properties of open sets and how they relate to the topology as a whole. As you continue to study and work with different examples, it will become clearer. Keep at it!