# Convergence in a Topological Space

# By Zehao Dong and Zihan Hu

# 1 Introduction

In Chapter 2, we studied the definition of sequence and the convergence of a sequence. Topological spaces provide a general framework for the study of convergence. However, instead of a distance function, we can think of the basic structure on a topological space as a collection of open sets.

Definition: In terms of open sets, a topological space is an ordered pair (*X, τ *), where *X *is a set and *τ *is a collection of subsets of *X*, satisfying the following axioms.

- The empty set and
*X*itself belongs to*τ*. - Any union of members of
*τ*belongs to*τ*. - The intersection of any finite number of members of
*τ*belongs to*τ*.

*Ex *: Let *X *be set such that *X *= {*a, b, c*}. Then let *τ *be a collection of the subsets of *X *such that *τ *= {ø*, *{*a, b*}*, *{*a, c*}*, X*}. Then, even though the first two properties are satisfied, the intersection of {*a, b*} and {*a, c*} is {*a*}, which is not in *τ *. Therefore, *τ *is not a topology of the set *X*.

# 2 Relating to Last Week’s Reading

There are many ways of defining a topology on R. The standard topology on R is generated by the open intervals. we can define a topology *τ *on *X *= R by defining that *T *⊆ R to be in *τ *if, for every point *x *∈ *T *, there exists an e* **> *0 such that (*x *− e*, **x *+ *e*) ⊆ *T *. This topology is defined as the **usual topology** on R. For example: Let *X *= R and let *τ *= {∅*, *(1*, *3)*, X*}.

- The subset {1, 2, 3} will not be in τ because it only contains the points 1, 2, and 3, but, for example, for point 1∈ T, there is not a value e > 0 so that (1 – e, 1 + e) is a subset of T. Hence, this T is not in τ.
- The subset (1, 2) will be in τ because for example, for x = 3/2 ∈ T, there exists an e = 1/4 so that (3/2 -1/4 , 3/2 + 1/4) = (5/4, 7/4) is a subset of T. And we need to prove that for every x ∈ T, there exists an e > 0 such that (x − e, x + e) ⊆ T to prove that T∈ τ.
- The subset [1, 2] will not be in τ for the similar reason as 1, for point 1∈ T, there is not a value e > 0 so that (1 – e, 1 + e) is a subset of T so this T is also not in τ.

This definition of usual topology can be used to explain convergence of a sequence and it is similar to the Theorem 2.2.3B in our reading last week.

By Theorem 2.2.3, a sequence (*x**n*) converges to a real number *x *if, for every positive number *e*, there exists an *N *∈ N such that whenever *n *≥ *N *it follows that |*x**n *− *x*| *< e*. This is an example of a distance function mentioned in our introduction, where |*x**n *− *x*| represents the distance between *x**n *and *x *on the line of R.

Theorem 2.2.3B is saying that a sequence (*x**n*) converges to *x *if, given any *e*-neighborhood *Ve*(*x*) of *x*, there exists a point in the sequence after which all of the terms are in *Ve*(*x*). In other words, every *s*-neighborhood contains all but a finite number of the terms of (*x**n*).

And to say a sequence (*x**n*) in *X *converges in the topology *τ *to an element *x *∈ *X *if, given any set *T *∈ *τ *such that *x *∈ *T *, the sequence (*x**n*) is eventually in *T *.

Here we can see that convergence in topology is similar with Theorem 2.2.3B from topological aspect. We can treat *Ve*(*x*) as a of a type of *T *∈ *τ *that contains the point *x *to which (*x**n*) converges. Let *T*1 = (*a, b*) ∈ *τ *be an open interval such that *x *∈ *T*1 and *T*2 = (*c, d*) ∈ *τ *be an open interval such that *x *∈ *T*2 with *a, b, c *and *d *arbitrary.

Let *τ *= {∅*, T*1*, T*2*, **X*}.

- The empty set and
*X*belongs to*τ*. - The union of members of
*τ*belongs to*τ*. ex.*T*1 ∪*T*2 ⊆*X*∈*τ*.

- The intersection of any finite number of members of
*τ*belongs to*τ*. ex.*T*1 ∩*T*2 =*x*∈*X*∈*τ*and*T*1 ∩*X*=*T*1 ∈*τ*.

So *τ *will be a topology because it satisfies all the three properties of topology mentioned above. And the sequence (*x**n*) in *X *converges in the topology *τ *to an element *x *∈ *X *if, given any set *T *∈ *τ *such that *x *∈ *T *, the sequence (*x**n*) is eventually in *T *.

# 3 Examples of Topologies

For *X *= R, we have two extreme examples:

- Trivial Topology:
*τ*= {ø*,*R}. - Discrete Topology:
*τ*= P(R) (i.e. The power set of R, which contains all possible subsets of R).

Notice that both topologies satisfy the properties mentioned before. Both topologies contains ø and the entire set *X *which equals to R.

For the trivial topology, the intersection of the two elements is ø, which is in *τ *, and the union of the two elements is R, which is also in *τ *.

For the discrete topology, since it contains all subsets of R, the intersection or union of any elements inside the power set of R will still be in *τ *.

# 4 Examples of a Sequence In R That Converges In Some Of the Topological Space

Before digging into examples, let us review the definition of convergence of a sequence in topological space.

Definition: Let (*X, T *) be a topological space and *x**n *∈ *X *a sequence. We say that the sequence *x**n *converges to *x*0 ∈ *X *if for every open set *U *⊆ *X *which contains *x*0 there exists an *n*0 ∈ N such that for all *n ≥* *n*0 the points *x**n *lies in *U *.

Then, consider the following three sequences.

*a**n*= 1/n.*b**n*=*n*.*c**n*= 1.

We know that in the usual topology, *an* converges to 0, *cn *converges to 1, while *bn* diverges. However, this is not the case in other topologies.

In trivial topology {ø, R}, the only open set that contains the limit of the three sequences is R. Since for all of the three sequences, xn ∈ R, ∀n ∈ N, all of them converge in the trivial topology.

In discrete topology, notice that T = {0} is in the topology, and since *an* converges to 0, by definition, because *an* never enters the set T (for all n, *an* ≠ 0), *an* does not satisfy the condition of being eventually inside every set T in τ which contains the point 0, so *an* fails to converge to 0 in the discrete topology. *bn* still diverges in the discrete topology. Because *cn *is contained in every subset in the discrete topology that contains 1, *cn** *converges in the discrete topology.

# References

https://en.wikipedia.org/wiki/Topologicalspace

This is a good prelude to (some) material we will discuss in Chapter 2. Thank you for sharing what you have learned, Zehao and Finnegan.

A bit of caution is necessary for students as they read through Section 2 of this post. There are two instances where the authors “construct” a topology τ (either as the collection τ = {∅, (1, 3), X} or the collection τ = {∅, T1, T2, X}), however they are implicitly discussing the USUAL topology, which cannot be constructed in this way. (In fact, the diligent reader will note that τ = {∅, T1, T2, X} is NOT a topology, because neither T1 ∪ T2, nor T1 ∩ T2 are necessarily elements of the topology as stated. They may be subsets of elements in the topology, but this does not make them elements of the topology themselves… more care is needed here.)

Rather, the usual topology contains ALL subsets of R within which every point has a e-neighborhood contained inside the set. There are uncountably infinitely many sets in this topology (it is more appropriate to consider it as slightly smaller than the power set, than to try to relate it to collection of 3 or 4 subsets). The goal in this discussion is not to confirm that the usual topology is indeed a topology (if you are curious about proving this, you need more time and a more general argument; parts of which we will see in Chapter 3 of the textbook), but rather to discuss how convergence in the usual topology relates to convergence as we’ve defined it in class via the distance function |x – y|.

To be precise, saying that (an) converges in the usual topology to a, means that given any open set T so that a is contained in T, the sequence (an) is eventually in T. Exploring the definition of open set, one will note that T open and a in T, means there must exist some e > 0 so that (a – e, a + e) is a subset of T. But (a – e, a + e) is ALSO an open set (not the same open set as T, generally, but still an open set), so convergence means that (an) is eventually in (a – e, a + e) as well… which is directly related to the definition of convergence we have seen in class.