Introduction to Topology

In this article I want to provide a quick introduction to topology. There are many books that are introductions to topology that are a couple inches thick so this can be thought of as more of a commercial for topology to see if you enjoy it. I want to introduce some of the ideas that are the most important and I think to most interesting. In this article I assume you already have a little set theory background including the ideas of unions, intersections, and compliments, and I assume you have a background with types of numbers like natural numbers, integers, and reals.

We are going to be talking about what is called point-set topology. Topology can be though of as the study of geometric properties that stay constant even when the space is bent and stretched. Point-set topology is actually more general and it can apply to situations that have nothing to do with geometry although it does generalize geometric type properties of situations beyond geometry. An example of this is to consider a square and a circle. A square can be bent and stretched into a circle, but it cannot be bent and stretched into a figure eight so a square and circle are topologically the same while a square and circle are not. This difference can be expressed entirely in terms of open sets which is why once you define open sets you have defined a topology. You can think of a topology as sort of like a parallel universe where the properties of geometry can be different. The properties of our universe that we live in would be R^3 which is three dimensional real space. For an introduction to topology you typically start with the properties of the real numbers R and then move on to the definition of a general topology. You then look at some different topologies and see of topological properties are different in each of them. In this article I am going to start with the topology of R and then move on to other topologies and compare different properties. I am using keyboard to type so I will use certain keyboard shorthands for mathematical symbols (I plan on publishing on a website later that will be able to use standard mathematical symbols.) Here are a list of some of the symbols I will use. I assume you are familiar with the ideas behind the following terms.

{} - The empty set

U - A universal set

A^c - The set theoretic compliment of a set

N - The set of natural numbers

Z - The set of integers

Q - The set of rational numbers

I - The set of irrational numbers

R - The set of real numbers

(a,b) - An open interval between a and b

[a,b] - A closed interval between a and b

union(A,B) - The set theoretic union of two sets

intersection(A,B) - The set theoretic intersection of two sets

union(Ai) - The union over an index i of all the Ai sets

intersection(Ai) - The intersection over an index i of all the Ai sets

The following are notions used in topology.

X - The space a topology is defined over

tau - The topology on a space

cl(S) - The closure of a set

int(S) - The interior of a set

First we start by analyzing the real numbers. The real numbers are everything on the number line. An epsilon neighborhood of a point p in the real numbers is the open interval (p-episolon,p+episolon). We define a set to be open if every point in it has an epsilon neighborhood in the set. You then can notice that every open neighborhood satisfies three properties. {} and R are open. The union of open sets is open. A finite compliment of open sets is open. It turns out we can define any topology by assuming these properties. For now lets study open sets in R.

Open Sets

Example - An open interval is an open set

Proof - Any point in an open interval will have a finite distance to the endpoints. Choose epsilon smaller than this and you have proved there exists an epsilon neighborhood contained in the set.

The proofs I use will be more informal since I want to aim this article more at explaining the concepts.

Example - R is an open set

Proof - For any point in R you can choose an arbitrary epsilon neighborhood and this will still be contained in R

Example - {} is an open set

Proof - There are no points in {} so it is true vacuously.

When we say something is true vacuously it means that you can say anything you want about something that does not exist. For example the statement "All unicorns are blue" is a true statement. You can think of this as the fact the there are no unicorns means that the statement the they a blue is not wrong. Weirdly, it turns our the statement "All unicorns are not blue" is also a true statement. Formally, the statement "all unicorns are blue" can be thought of as "If there exists a unicorn it is blue". Likewise the statement "All unicorns are not blue" can be thought of as "If there exists a unicorn it is not blue." The contrapositives of the statements are "If a unicorn is not blue than it does not exist" and "If a unicorn is blue than it does not exist." The statements are both true because unicorns don't exist anyway therefor since the contrapositives are true the original statements are also true. Thus saying that all points in the empty set have an epsilon neighborhood contained in the set is like saying all unicorns are blue and so it is automatically true.

Example - (0,infinity) is an open set

Proof - Any point in the interval must have some finite distance to 0. Choose an epsilon less than this and the epsilon neighborhood must be contained in the set.

It actually turns out in the real numbers that every open set can be written as a union of open intervals. We would say that open intervals form a base for the standard topology on the reals.

Closed sets

The next idea is a closed set. A very simple definition of a closed set is that a set is closed if its compliment is open. Symbolically

A is closed if and only if A^c is open.

Example - A closed interval is closed

Proof - A closed interval is of the form [a,b]. The compliment is union((-infinity,a),(b,infinity)). From before we saw sets of this form are open.

Example - R is closed

Proof - R^c is {} which is open

Example - {} is closed

Proof - {}^c is R which is open

Sets which are both open and closed are called clopen. This shows that open is not the opposite of closed. There are also sets which are neither open nor closed like [0,1).

Limit Point

A limit point of a set is a point where each epsilon neighborhood intersects the set in at least one other point.

Example - 0.5 is a limit point of (0,1)

Proof - Any epsilon neighborhood will either be contained in or contain the set so the intersection and since both are nonempty the intersection must be nonempty.

Example - 1 is a limit point of (0,1)

Proof - Any epsilon neighborhood will contain points less than 1 and greater than 0 and thus contained in the interval. Therefor the intersection will be nonempty.

Notice that in the second case 1 is a limit point even though it is not a value of the original set.

NonExample - 5 is not a limit point of Z

Proof - Let epsilon be 0.5. Then that particular epsilon neighborhood contains no other points in the set besides 5 itself therefor the intersection is empty.

Notice that 5 is not a limit point even though it is a value of the original set.

Example - 0 is a limit point of the set {1/n: n contained in N}

Proof - The set is the set of the reciprocal natural numbers {1, 1/2, 1/3, 1/4,...}. Every positive number is going to be larger than some value in this set. Thus for any epsilon one of the points in this set will be contained in the epsilon neighborhood and thus the intersection will be nonempty.

Notice in this case that 0 is the only limit point of this set.

It turns out that limit points provide another definition of a closed set. A set is closed if and only if it contains all its limit points. You can look at the examples of closed sets and verify that they satisfy this criteria.

Closure

The closure of a set is the set is the disjoint union of the limit points and the isolated points. Isolated points are defined as points that are contained in the set and are not limit points. These are the points that exist on their own like 5 in Z. You can also define the closure as the union of the set of limit points of a set with the original set although this will not in general be a disjoint union. We refer to the closure of a set as cl(S). The closure is actually an operator and more formally it is a set theoretic function that transforms sets to other sets.

Example - cl((0,1))=[0,1]

Proof - As we saw before points contained in the set are limit points and 0 and 1 are also limit points so together this makes the closed interval [0,1].

Example - cl([0,1])=[0,1]

Proof - [0,1] is already a closed set and you can see that the closure of any closed set will be itself since every point in a closed set is a limit point and every limit point of a closed set is already in the set.

Example - cl({1/n: n is contained in N}=union({1/n: n is contained in N},0)

Proof - We proved before that 0 is a limit point of this set. To show it is the only limit point notice for any point positive number you can find a finite distance to the closets other point. Take epsilon to be smaller than this and you will have an epsilon neighborhood that has an empty intersection with the set. For negative numbers take epsilon to be any number with absolute value smaller than the original number any you will have an epsilon neighborhood with empty intersection with the set.

Notice for the above examples the closure could be though of as the simplest way to make the set closed. It turns out that the cl(S) is the smallest closed set that contains S. Thus any closed set which contains S either is equal to cl(S) or contains it.

Interior

Interior points are defined to be points that have an epsilon neighborhood contained entirely in the set. These are points that fit comfortably in the set and there is no space around them. The interior of a set is the set of all interior points. We write the interior of S and int(S) which is also an operator on S.

Example - int((0,1))=(0,1)

Proof - All points contained in the set as we have shown before are limit points. 0 and 1 are limit points, but are not in the set.

Example - int([0,1])=(0,1)

Proof - Every point inside the set is an interior point, but 0 and 1 are not since every epsilon neighborhood of each intersects points outside the set.

Example - int({1/n: n contained in N})={}

Proof - Every point of this set is an isolated point and thus every point must have an epsilon neighborhood that does not intersect other points in the set.

The interior can be though of as the simplest way to make the set open. It is true that for an open set, a point is an interior point if and only if it is in the set. Therefor the interior of an open set is the original set which or symbolically if S is open int(S)=S. Also int(S) is the largest open set that is contained in S. Thus any open set contained in S either equals int(S) or is contained in it.

Connectedness

We say that a set is connected if it is impossible to find two open sets that contain it with empty intersection. This means that if there are two parts of the set such that you can find two open sets that contain the parts and the intersection of the two open sets is not empty they original set is not connected. Intuitively, a set is connected if it is in one piece. Thus the set union((1,2),(4,5)) is not connected. An example of two open sets that break it up would be (0,3) and (3,6).

Point Set Topology

Now let's move on to the definition of topology. The idea is that we generalize the notion of a collection of open set to any set. We can choose any collection of open sets we want, but they must satisfy certain properties. The topology is sort of like its own universe where properties might be different. Thus a set that is connected in one topology might not be connected in another topology and so on. We use X to represent the topology the set is defined on and we use tau to represent the collection of open sets. X is also the universal set U so union(S,S^c)=X. Tau is a set of subsets of X which are defined to be open. For example consider the set X={a,b,c,d,e} and tau={{},{a,b},{c,d,e},{a,b,c,d,e}}. You can verify that this satisfies the requirements of a topology. {} and X are both in the set (they are the first and last entries). Any union of members of the topology is still in the topology. Any intersection of members of the topology is still in the topology. Let's go over the properties of a general topology.

Open Sets

The open sets are simply the sets defined in the topology. All other topological properties can be expressed in terms of the opens sets and set theoretical properties of X itself. In our example the open sets are {}, {a,b}, {c,d,e}, and {a,b,c,d,e}. The open sets must satisfy the properties

{} and X are open sets

Any union of open sets is open

Any finite intersection of open sets is open

There are the axioms that define a topological space.

Closed Sets

The closed sets can still be thought of as the compliments of open sets. They can also be thought of as the union of the original set and the limit points. In the above example the closed sets are {}^c={a,b,c,d,e}, {a,b}^c={c,d,e}, {c,d,e}^c={a,b}, {a,b,c,d,e}^c={}. In this particular every closed set was also open and thus every set is clopen. It turns out that the whole topology will be connected if {} and X are the only clopen sets. In general a clopen set is a maximal connected component of X.

Limit Points

The definition of a limit point is the same.

Example - In the above topology a is a limit point of {a,b}

Proof - In order to be a limit point every open set containing a must have a nonempty intersection with {a,b}. There are exactly two open sets containing a namely {a,b} and {a,b,c,d,e}. We have intersect({a,b},{a,b})={a,b} is nonempty and intersect({a,b},{a,b,c,d,e})={a,b} is nonempty.

Limit points can still be thought of as being close to a set, but this definition becomes more ambiguous since space doesn't really exist in some examples like the one above.

Closure

The definition of closure is also the same

Example - cl({a})={a,b}

Proof - Every open set containing b has a nonempty intersection with {a}. The two open sets containing b are {a,b} and {a,b,c,d,e}. We have intersect({a},{a,b})={a} which is nonempty and intersect({a},{a,b,c,d,e})={a} which is nonempty. Thus b is a limit point of a. The closure is the union of the original set {a} with the limit points of a {b} so it is {a,b}.

An easier way to do this is to simply find the smallest closed set containing a.

Interior

The definition of interior is going to be the same.

Example - int({a,b,c})={a,b}

Proof - Doing it the easy was notice that {a,b} is the largest open set contained in {a,b,c}.

Note that this will never be ambiguous. If there was ever a situation where there were two maximal open sets contained in {a,b,c} that were distinct you would be working with something that is not a topology.

Connectedness

The definition is the same. You can also locate maximal connected sets by finding clopen sets.

Example - {a} and {c} are disconnected

Proof - {a,b} is an open set containing {a} as a subset. {c,d,e} is an open set containing {c} as a subset. Also intersect({a,b},{c,d,e})={} so {a} and {c} are disconnected.

Let's consider some other important examples of topologies and quickly go through the properties

The discrete topology

Open Sets

This topology is simply the power set of X. The open sets are defined to be every subset of X. For example the discrete topology on X={1,2,3} is tau={{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Closed Sets

It is easy to see that the closed sets will simply be all the open sets. The compliment of any subset is another subset and so every set is clopen.

Limit Points

A point is a limit point of a set S in the discrete topology if and only if it is contained in S. You can see this because every open set containing to point must have a nonempty intersection with S, but because every set is open every set containing the point must have a nonempty intersection with S which is only possible if the point is in S.

Closure

The closure of a set in the discrete topology is the original set. Symbolically cl(S)=S. The easy way to prove this is to note that the smallest closed set containing a set is the set itself(every set is closed).

Interior

The interior of a set in the discrete topology is the original set. Symbolically, int(S)=S. The easy way to prove this is to note that the largest open set contained in a set is the set itself.

Connectedness

In the discrete topology every two sets that are distinct are disconnected (this is where the name discrete comes from). The easy way to see this is that every set is clopen.

The indiscrete topology

Open Sets

The only open sets in the indiscrete topology are {} and X no matter what the set X is.

Closed Sets

{} and X are also the only closed sets since {}^c=X and X^c={}. Thus {} and X are clopen.

Limit points

The only set that has any limit points is X. Every point is a limit point of X since every open set (X is the only one) containing the point has a nonempty intersection with X.

Closure

The closure of any set is X. The easy way to see this is that X is the smallest closed set containing the set. Symbolically cl(S)=X

Interior

The interior of any set is {}. The easy way to see this is that {} is the largest open set contained in the set. Symbolically int(S)={}.

Connectedness

Every point in the set is connected. This is because you can't find two open sets containing different points with a nonempty intersection. The only open sets in the entire topology are {} and X and they have an empty intersection. The indiscrete topology has its name because it can be though of as the counterpart of the discrete topology.

The lower limit topology

This is a topology on R. This one is a little more geometrical because we are working with R again. Recall that a base is a collection of sets such that the union of the set can form any set in the topology. This is similar to the notion of base in linear algebra. The open intervals of the form (a,b) are an example of a base for the standard topology on R. The lower limit topology is defined by a base of intervals of the form [a,b). You can also define the upper limit topology with a base of (a,b] intervals.

Open Sets

The open sets are unions of intervals of the form [a,b).

Example - union([1,2), [4,8), [9,15), [18,20)) is an open set in the topology

Proof - Follows from the definition

Intervals that go to infinity on the right or left are also open and you can see that they satisfy the above definition.

Closed Sets

The closed sets are unions of intervals of the form (a,b]. To see this note that the compliment of an open set must be an interval (if intervals that go to infinity are considered intervals). The upper part of any component must be open and the lower part must be closed.

Limit Points

Limit points of a set in the lower limit topology only exist if the set approaches from the right.

Closure

The closure of a set will typically be of the form (a,b] or a union of these.

Interior

The interior of a set will be of the form [a,b) or a union of these.

Connectedness

The only connected set in the lower limit topology are single points. Intervals are not connected

Example - (1,3) is not connected in the lower limit topology

Proof - (1,3) can be partitioned into [0,2) and [2,4) which contain the previous sets and have an empty intersection.

In general a fairly good way to learn topology is to make a list of properties and compare these properties for different topologies. Once you have point set topology down there are a couple ways you can go further into topology.

Continuous Functions

You may be used to a function that is continuous in real analysis as a function which doesn't have any gaps. This is the real analysis version and is a special case of functions from R to R. In topology a continuous function is a function that maps open sets in one space to open sets in another space. When both spaces are R this is equivalent to the definition above. If you remember all of the topological properties of a space are determined by its open sets so thus a continuous function doesn't change the basic nature of a topological space. A homeomorphism is a function which is continuous and has a continuous inverse so it relates two topological spaces that differ in only a superficial way. You might remember the idea of stretching and bending a square into a circle. This would be an example of a homeomorphism. The idea of stretching and bending can sometimes lead you wrong, for example the letter x (its shape) is not homeomorphic to the letter y because x has 4 tails and y has 3 even though they might look like you could stretch one into the other. There are actually two different ways to look at many topological properties. For example, if you want to show that union((0,1),(4,5)) is topologically distinct from (0,4). You could show using open sets in R that the first is not connected while the other is. In this case you are using the topology of R to show that the two subsets of it have different properties. You could also show that no homeomorphism exists between the two sets. In this case you are considering them as separate topological spaces and showing they cannot have the same topological properties.

Separation

The separation axioms are an important part of topology. They provide different ways of measuring how far apart two points are by considering different conditions. The separation axioms are often called T0, T1, and so on to T6.

Metric Spaces

A metric space provides a way to measure the distance between points. It some of the proofs above I informally referred to the distance between points which takes advantage of a metric on R. On R the distance between two points a and b is simply b-a if b>a. Otherwise a-b will be negative to to get rid of this issue we use the absolute value |b-a|. This is called the Euclidean Metric on R. On R^2 we would have sets of points p1=(a1,b1) (note this is a coordinate point, not an interval) and p2=(a2,b2) so the distance between p1 and p2 would be sqrt((b1-a1)^2+(b2-a2)^2). This is the Euclidean Metric on R^2. An example of a different metric would be the taxicab metric where the distance between p1 and p2 is defined to be |(b1-a1)+(b2-a2)|. In a city the taxicab metric would be the distance you would have to go between points going up a street and across an avenue as opposed to going as the crow flies. You can form a topology out of a metric space by having open sets formed using open balls of a given distance (using the metric) from points. All metric spaces are topologies, but not all topologies are metric spaces. Topologies that can be given a metric are called metrizable.