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 ...?
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

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