From Armstrong:

On page 13, we have definition **1.3**:

Given a set and for each point in a nonempty collection of subsets of , called

neighborhoods of. These neighborhoods are required to satisfy four axioms:

- lies in each of its neighborhoods.
- The intersection of two neighborhoods of is also a neighborhood of .
- If is a neighborhood of and is a subset of which contains , then is a neighborhood of $latex x.
- $If is a neighborhood of and denotes the set , then is a neighborhood of . (The set is called the
interiorof .)In which case we can call (along with the specified neighborhoods) a

topological space.

Whew! That’s quite a bit, but it simply gives us a firm standing for the intuitive idea of a *neighborhood*, and then declares that when we have a set such that every point has at least one neighborhood (the “nonempty collection” requirement above), then we have a topological space.

Does it seem to you that it might be a bit difficult to verify that we have a topological space? It’s not enough that we verify the axioms for some point in , we have to do so for *every* point! Perhaps there is an easier way…

Enter definition **2.1** on page 28:

A topology on a set is a nonempty collection of subsets of called

open setssuch that: any union of open sets is open; anyfiniteintersection of open sets is open; and both and are open. A set together with a topology on it is called atopological space.

Well, that *seems* a little simpler, but does it describe the same structure as the first definition? (Hint: yes, it does.) If we read carefully the beginning of chapter 2 (pp. 27-8), Armstrong shows the logical equivalence in both directions.

To show that **1.3** implies **2.4**, we just describe a set as *open* if it is a neighborhood of each point it contains. The properties of **2.4** then follow from **1.3** fairly easily.

To go the other way, we take any point and then say that a subset of is a *neighborhood of * if there exists an open set such that . It is then a matter of a little work to show that all of the axioms of neighborhoods are then satisfied.

So- that’s it for the moment. Have a great weekend, and we’ll see how our first classes go on Monday!

ex animo-

Felicis

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