A summary of Chapter 3 of the book “Topology without Tears” by Sidney Morris that I’m reading.

Limit Points

Definition. Let A be subset of a topo space . A point is called a limit point of A (aka accumulation point or cluster point) if that contains x contain a point of A other than x. In other words,

Theorem. is a closed set in the topo space iff A contains all of its limit points. ▶ Proof

Suppose that A is closed and but is a limit point of A. Therefore, that contains x contain a point of A other than x. However, is an open set containing x but does not intersect A. Hence x is not a limit point, which contradicts to the assumption. Hence if A is closed then it contains all of its limit points.

Suppose that A contains all of its limit points. For any , x is not a limit point, so such that and , i.e. . We can write X-A as an infinite union of open sets, hence X-A is open and A is closed.

Note that if doesn’t guarantee it is a limit point. For example, if A is a finite subset of the topology , no point in A is a limit point, neither is any other point in is a limit point. Because for every , you can always find an interval containing x but intersect trivially.

Another example is that in the discrete topology , no point is a limit point of any set. Recall that the basis of the discrete topology is the set of singleton sets. For any point x and a subset A of X, is an open set containing x but .

Closure

Theorem. Let A’ be the set of all limit points of A in the topo space . Then is a closed set and it is called the closure of A.

This gives us some conclusions:

  •   is the smallest closed set containing A.
  • Every closed set containing A must also contain .
  •   is the intersection of all closed sets containing A.
  • A is closed iff A is a closure of itself.

Examples:

  • The closure of in is ▶ Reveal

It has been proven that is not a closed set, but is a closed set, and also the smallest closed set containing A, hence it is the closure of A.

  • The closure of in is . ▶ Proof

Suppose that , i.e. (an open set) we have such that . However this interval also contains rational numbers, which contradicts to . Hence .

  • The closure of in is because it is a closed set.
  • The closure of in is because any interval contains infinitely many irrational numbers, hence intersect non-trivially.

Definition. Let A be a subset of a topo space . We say A is dense in X (aka everywhere dense) if .

Theorem. A is dense iff every non-empty open set intersects A non-trivially. ▶ Proof

Suppose that A is dense, i.e. , but there exists a non-empty open set U that , so is not a limit point of A. This contradicts to the assumption that every point in X is a limit point. Hence every non-empty open set intersects A non-trivially.

Suppose that every non-empty open set intersects A non-trivially. For any point , any open set U containing x is surely non-empty, so . Since , we have , hence x is a limit point. Now we have that all points in X-A are limit points of A. The closure of A is hence .

Theorem. . ▶ Proof

Let be a limit point of . We have that open and containing x:

That means x is a limit point of A and a limit point of B. .

Now we shall prove that .

The theorem has been proven.

An example of is when and . The LHS is while the RHS is .

Theorem. Let S be a dense subset of a topo space . For every open set , . ▶ Proof

The proposition is obviously true when , hence we only take care of the case below.

First,

We need to prove .

Let be a limit point of . We need to prove that , i.e. is a limit point of or belongs to . Anyway, we have that for any open set A containing x, .

  • If is a limit point of , we also have . Moreover, since is dense and is open, we have . This implies that . Hence is a limit point of . .
  • If is not a limit point of , for sure we have (because S is dense, if x was not in S then x would be a limit point). That means there exists an open set containing such that . Moreover, we already know , we can conclude . Since is open, we also have , which implies . We already know , hence , which leads to , and finally . This means .

So, if is a limit point of , .

Now, what if is not a limit point of ? In that case, we’re sure , otherwise is one of the limit points added to the closure. Since is not a limit point, there exists an open set such that . Furthermore, we know that and , therefore . Since is dense and are open sets, and are non-trivial. This leads to . Hence , which implies , and finally .

Hence .

Hence .

Neighborhood

Definition. Let a topo space, N a subset of X, and p a point in N. N is called a neighborhood of p if there exists some open set U such that .

One can have an alternative definition of limit point from this: A point x is a limit point of A iff every neighborhood of x intersect A at a point other than x.

Connectedness

Given a set of real numbers. As you may have known, if there exists such that is greater than any other numbers of S then b is called the greatest element. S is said to be bounded above if there exists a real number c such that c is greater than any element in S. We call c an upper bound of S. The least upper bound is called supremium. Similarly, the greatest lower bound is called infimum.

Lemma. Let and S is bounded above with p being its supremum. If S is closed, then .

Theorem. The only clopen subsets of are and .

Definition. Let a topo space. It is said to be connected if the only clopen sets are X and .

Remark. Let a topo space. It is said to be disconnected iff there exists a non-empty set A different from X such that A and X-A are open.

The notion of connectedness is very important and we shall discuss in the next posts.

Reference sources

  1. Sidney Morris - “Topology Without Tears”

Special thanks to Tran Hoang Bao Linh for giving some nice examples in this post.