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

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