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Mar10-11, 11:53 PM
P: 31
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) [tex]\emptyset \in O[/tex] and [tex]E \in O[/tex]

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 [itex]\epsilon>0[/itex] 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?
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