Informally, a topology is a kind of “arrangement” or “organization” that we put on a set. One can think of an analogy with an army, which is made up of soldiers organized into squads, which are in turn organized into platoons, and so forth. Topology accomplishes this by organizing subsets of a set into “open sets” and “closed sets”. We quote here the rigorous definition of a topology following the book Topology by James R. Munkres:
A topology on a set is a collection of subsets of having the following properties:
(1) and are in .
(2) The union of the elements of any subcollection of is in .
(3) The intersection of the elements of any finite subcollection of is in .
A set for which a topology has been specified is called a topological space.
From the same book, we have the following definition of an open set:
If is a topological space with a topology , we say that a subset of is an open set of if belongs to the collection .
We also have the following definition of a closed set:
A subset of a topological space is said to be closed if the set is open.
We note that the notation here refers to the complement of in , i.e., the set is the set of all elements of the set that are not also elements of the set .
Our definition of open sets and closed sets has some results that seem rather weird at first glance. By definition, both the entire set and the empty set are open. But is just , which is open; therefore, by our definition of closed sets, is closed. Similarly, since is the empty set , which is again open, we find that the entire set is also closed. Therefore, the sets and are both open and closed!
This only seems paradoxical because we are used to thinking of the words open and closed as being opposites; such may be the case in real life, but for our purposes these words are merely terminology that we use in order to organize our set; therefore, it should not be troubling for us to find a set both closed and open (some refer to such a set as a “clopen” set). There also exist examples of sets in some topologies being neither closed nor open.
Now we show one example of putting a topology on a set. We consider the set with two elements, which we shall refer to as and . This set has the following subsets:
We shall now put a topology on this set. By the definition of a topology, the subsets , which is the entire set, along with the empty set have to be open. By the result we discovered earlier, they must also both be closed. We now have a choice of what to do with the remaining subsets, and . If we declare them to both be open, then all the subsets are open sets. We call a topology where all subsets are open the discrete topology. It so happens that if we do this, both and will also be closed by the definition of topology. So putting the discrete topology on this set with two elements makes all subsets both open and closed.
We can also not declare anything on the two sets and ; this will make them neither open nor closed, and only the entire set and the empty set are declared open (they also happen to be closed). Such a topology where only the entire set and empty set are declared to be open (which they are forced to be, by definition) is called the trivial topology.
Finally, we can declare just one of and to be open. There are two different ways of doing this; we can declare either to be open, in which case will be closed, or we can declare to be open, in which case will be closed. A two element set where one of the one-element subset is declared to be open, rendering the other one-element subset closed, is called a Sierpinski space.
We tackle one more example. Consider the set of all real numbers , also called the real line. We will put a topology on the real line, but first there is one more concept that we need to define. Let and be two real numbers, where is greater than (also written, of course, as ). The set of all real numbers which are greater than but less than is denoted by . Note that and themselves are not included in the set . We call sets such as these open intervals. If instead we consider the set of real numbers greater than or equal to but less than or equal to , then we write , and both and are now included in . Such sets are called closed intervals.
We now go back to putting a topology on the real line. As may be suggested by the naming, we now declare that all sets that are unions of open intervals, including of course the open intervals themselves, to be the open sets of our topology. This will make the closed intervals, including the sets consisting of only one real number, into closed sets. We will not explain why in this post, but it always goes back to the definitions of topology, open set, and closed set. This topology that we have defined on the real line is called its standard topology.
Now that we know the concept of an open interval, there is another related concept that we will introduce in this post. We stay in the context of real numbers and the real line. Let be a real number, and let be a positive real number. The open interval is an example of what we call a neighborhood of . It consists of all real numbers whose difference from is less than , or we may think of it them as being less than a distance away from . This motivates the terminology of “neighborhood”, even though can be as small or as big as we want.
The concept of a neighborhood plays a big role in a very common kind of topology called a metric topology. It plays a big role in the modern foundations of calculus and geometry.
Recall that a function is a mapping between sets, in the sense that it assigns to every element in a set called its domain an element of another set called its range. As per our definitions above, a topological space is just a set for which a topology is specified, so we can talk about functions between topological spaces. The topologies on the sets involved will allow us to define an important kind of function between topological spaces, called a continuous function. Once more we refer to the book of Munkres:
Let and be topological spaces. A function is said to be continuous if for each open subset of , the set is an open subset of .
Recall that is the set of all points of for which ; it is empty if does not intersect the image set of .
Continuity of a function depends not only upon the function itself, but also on the topologies specified for its domain and range. If we wish to emphasize this fact, we can say that is continuous relative to specific topologies on and .
A continuous function with a continuous inverse is called a homeomorphism.
This is the most basic definition of continuity of a function. However, depending on the topologies on the domain and range, there may be several equivalent definitions, all deriving from this one most basic definition, that will shed light on certain concepts of importance for the topological spaces that we are studying. We state here one important equivalent definition for the case of functions from real numbers to real numbers, with the set of real numbers equipped with the standard topology discussed earlier.
A function is said to be continuous if for any two real numbers and and a positive real number , there exists another positive real number such that whenever is less than , then it is guaranteed that is less than . The notation stands for the absolute value of ; if is greater than then it is simply equal to , but if is greater than then it is instead equal to . The same applies to . If is greater than then it is equal to but if is greater than then it is equal to .
The idea that this definition of continuity is supposed to communicate, is that we can always produce as small a change as we want in the “output” of the function as long as we make a change in the “input” that is sufficiently small enough. We can think of functions that are not continuous as having abrupt “jumps” such that even if we make the smallest of changes in the input we still cannot make the output change slowly enough with respect to this change in the input.
It is important to remind ourselves, once again, that this latter definition of continuity follows from the one most basic definition of continuity we have defined earlier, we have simply specialized it to the case where the domain and range is the set of real numbers, and we have equipped this set with its standard topology. We have not discussed explicitly how exactly to relate the two definitions here, but the inquisitive reader can find it and much more in the book of Munkres.
Topology by James R. Munkres