As shown in the textbook all compact sets in the space of real numbers must be closed and bounded (Abbot 96). However, topological spaces go far beyond just a set that is a subset of the reals. This brings to our attention the idea that the proof for sets being compact if and only if they are closed and bounded may not hold true for other topological spaces. So an obvious question presents itself:

Are all compact sets closed and bounded?

To fully understand and comprehend this question we need a full understanding of what it means to be compact, closed, and bounded.

Compact: 

• First, for a set to be compact, every sequence in the set must have a subsequence that converges to a limit that is also contained in the set.
• Further, if a set is a compact set it means that given any open cover we can find a finite subcover that covers the set (Abbot 96).

Bounded: A set is bounded if there is a bound that the all values in the set have an absolute value less than or equal to that bound.

Closed:

• First, a set is closed if it contains all of its limit points.
• Further, a set is closed if and only if the set’s complement is open (Abbot 90).

With everything defined it is now reasonable to address answers to the question.

Claim: There exist compact sets which are not closed.

The easiest way to demonstrate this will be to find a topological space within which there is a compact set that is not closed. Take the general set and a potential topology on  = . First, we need to show that  is actually a topology. To be a topology it needs to satisfy three conditions from the definition (which can be found on mathworld.wolfram.com. First, the empty set and the set itself must be in the topology which holds true here because and . Next, the union of any combination of sets in the topology must be in the topology which is true because , , , and  so all of the possible combinations of the unions of the topology are in the topology. Finally, the topology must contain the intersection of any finite combination of the sets in the topology which is again true because , , , and  so all possible combinations of intersections are in the topology. Since these three conditions are met it follows that is in fact a topology.

The next step is to show that there is a set in the topology that is compact but not closed. Choose the set . This set is compact because there is only one sub-sequence in {a} which obviously must converge to a so every sequence has a sub-sequence that converges to a limit inside the set, hence it is compact. However, the set is not closed because the complement of all closed sets must be open, and the complement  is not in the topology, hence not open. Since,  is not open, it follows that {a} is not closed. So this set and topology provide a counterexample that not all compact sets need to be closed.

The set I used to prove that a compact set does not have to be closed is known as a Sierpinski Space. The Sierpinski Space is a finite topological space with two points, only one of which is closed, which is key in proving that not all compact sets are closed outside of the reals. However, this simple topological space is crucial for a lot of general topology. For example, it is the smallest topological space that is neither trivial or discrete that can be constructed. Trivial means that the space contains more than just the empty set and the whole space while discrete means that it is a space that forms a discontinuous sequence. Also, it contains qualities that often contradict standard properties of sets in , such as being compact and not closed but also in more advanced topology. With its very many unique qualities it is an extremely useful tool in proving theorems for general topology. While most of the uses of this space are advanced it is interesting to note that what seems to be a simple space is an extremely powerful tool for mathematics.

Now to address boundedness look at the diagram above. To demonstrate that a compact set need not be bounded focus on the two outer most circles, the topological space and metric spaces. A metric space is a space where the distance between any two elements is defined (more details here). is a metric space with distance between members being |a-b| where . As you can see in the diagram above not all topological spaces are metric spaces. This means that there are topological spaces where the distance between two members of the set are undefined. As mentioned earlier a set is bounded if there is a bound that the all values in the set have an absolute value less than or equal to that bound which means such that the distance between elements is less than M,  when x and y are elements of the set in question. However, as stated before the distance between members of a topological space that is not metricized is undefined so you cannot say that because d(x,y) is undefined and therefore you cannot say that any set in the space is bounded. Since, is a metric spaces and we are working with topological spaces that are not metric Theorem 3.3.4 (Characterization of Compactness in ) no longer applies. So a compact set does not need to be closed and bounded.

Claim: There exist compact sets which are not bounded.

One example of a topological space within which there are compact, unbounded sets, is the Zariski Topology. A Zariski topology “is a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.”

A slightly more understandable example is the cofinite topology on , wherein open sets are defined as those subsets of  that can be written as for some finite . Recalling the definition of compactness, suppose  is an open cover for some set . Then the first set contains all but at most finitely many points of the set C. Then we can choose a set from the original covering that contains , the first of these points missed by . Then another set must cover the second of these points , etcetera. Continuing this process for all the points missed by and you will get a finite subcover of the original open cover. Hence the set is compact. Notice that no restriction was placed on the set , so we have actually just shown that EVERY subset is compact in this cofinite topology. Hence, there is are unbounded sets C which are compact.

In class and in chapter 3 we deal entirely with sets on the reals. However, as shown by the example above in more general terms the same rules do not apply. As shown in the reals one of the defining factors of a compact set is that it must be closed and bounded. The fact that a compact set is closed and bounded in the reals is known as the Heine-Borel theorem. The fact that there is a very well-known and important theorem that only holds true for the reals demonstrates that studying properties in the reals is not only unique but also very important to the field of mathematics which is why there is a whole course called Real Analysis.

Sources:

http://math.stackexchange.com/questions/239998/compact-sets-are-closed

https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space

Abbott, Stephen. “Basic Topology of ℝ.” Understanding Analysis. New York: Springer, 2015. 85-109. Print.