Introduction to Topology – Part 1

Hi! These next posts will begin an introduction to topology, explaining concepts, examples, and problems as I work through James Munkres’ Topology book. We introduced a topological space in our article Basic Continuity, so the beginning of this should be a bit of review for those of you who have read that article. As we begin our adventure, the questions arise:

“How did the field come about?” and “What is a topology?”

To address the first question, topology came about from the study of continuous functions on the real line and euclidean space(you should check out our article Basic Continuity for more information on that). The field then developed under geometry and set theory, analyzing the properties of spaces under continuous deformations.

For the second question,

topology on a set X  is a collection of subsets of X ,\mathscr{T} , with the following three properties:

  1. \emptyset and X  are in \mathscr{T} .
  2. The arbitrary union of elements in \mathscr{T}  will yield another element in \mathscr{T} .
  3. The finite intersection of elements in \mathscr{T}  will yield another element in \mathscr{T}

We say U , which is a subset of X , is an open set of X  if U belongs to the collection \mathscr{T} .

Using this definition, we can say that X  and \emptyset are both open. We can also say that the arbitrary union of open sets is open and that finite intersections of open sets are open.

So now that we know the definition of a topology, what are some examples?

The collection of all subsets of X  is a topology on X : one can easily check that it follows the three properties required to satisfy our definition. This topology is called the discrete topology.

Similarly, the collection containing only X  and \emptyset forms a topology. Once again, it can be easily shown that this collection satisfies the three needed properties of a topology. This topology is called the indiscrete or trivial topology.

Lets try to look at a slightly more complicated example.

Let \mathscr{T}_{f} be the collection of all subsets U  of X  such that X  – U  (the complement of U  in X ) either is finite or is X . Then \mathscr{T}_{f} is a topology on X , called the finite complement topology. The proof that this is a topology is less intuitive than the previous two. So why not show this ourselves?

Proof: We will show that the three properties required for a topology are present in our collection.

  1. Since X  – X  is finite and X  – \emptyset is all of X , we know that both X  and \emptyset are in the collection.
  2. We also need to show that the arbitrary union of elements in \mathscr{T}_{f} forms another element in \mathscr{T}_{f} . We can state, using DeMorgan’s Law, that

 X –  \bigcup (U_{\alpha}) \bigcap \limits_{i=1}^{n} (X - U_{i})

           We know that the finite intersection of finite sets is finite,

           meaning that

X  –  \bigcup (U_{\alpha})  is also finite.

Applying our definition above, \bigcup (U_{\alpha})  is in our topology

      3.  Now we must prove that the finite intersection of elements            in \mathscr{T}_{f}  forms another element in \mathscr{T}_{f} . We can write the                          following equation for arbitrary n, once again using                          DeMorgan’s Law:

     \bigcup \limits_{i=1}^{n} (X - U_i)  = X – \bigcap \limits_{i=1}^{n} U_i .

          Since the union of finite sets is finite, the resulting

X \bigcap \limits_{i=1}^{n} U_i  is also finite.

          This means that \bigcap \limits_{i=1}^{n} U_i  is in our topology.

Now that we’ve gotten the general feeling for a topology from these examples, we need a way to refer to these topologies. For the next post in this series, we will look at something called a basis. This basis for a topology will allow us to define a topology without listing out all the subsets in the space. Similar to the generators of group theory, these will allow us to create topological spaces from a subset of its elements.

N.K. (J.T.)


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