Why would one want to generalize notions such as convergence and continuity to a setting even more abstract than metric spaces?
Topology deals with the relative position of objects to each other and their features. It is not about their concrete length, volume, and so on. Hence, topological features will not change if continuous transformations are applied to these objects. That is, topological features are preserved under stretching, squeezing, bending, and so on but they are not preserved under non-continuous transformations such as tearing apart, cutting and so on. That is, objects such as a circle, a rectangle and a triangle are from a topological point of view homeomorphic even though the shapes are geometrically rather different.
What features are therefore of interest such that it is worth studying topology?
Assume that is a closed curve, i.e. a circle, a rectangle, a triangle or something like we can see in the graph above. As long as we transform a shape in a continuous fashion, the relative positions of the points and will be similar: for instance, points that have been inside of will still be inside after the continuous transformation. Points that have been on the boundary will still be on the boundary.
Hence, the generalization of continuity and the concept of convergence are the two most characterizing features in topological spaces.
However, convergence and continuity in a metric space were based on a notion of a distance function for points . Set-theoretic topology generalizes the features of topological metric space and ought to be based on an axiomatized notion of “closeness”.
This post is based on the literature  to .
Let — short — be a topological space in this post.
Topology on a Set
The term “topology on a set” is based on an axiomatic description of so-called “open sets” with respect to some set-theoretic operators. It will turn out, that a topology is a set that has just enough structure to meaningful speak of convergence and continuous functions on it.
Definition 1.1 (Topological Space)
A topological space is a pair , where is a set and is a family of subsets that satisfies
(ii) if for an arbitrary index set ;
(iii) if for a finite index set .
The following video provides a rather unorthodox way of thinking about a topology. However, it might help to get a heuristic for topological spaces. It also mentions the connection between metrics and topologies.
Example 1.1 (Topologies)
(a) Let be a set and then is the so-called trivial, chaotic or indiscrete topology.
(b) The power set of a set is the so-called discrete topology. In this topology every subset is open.
(c) There are four topologies on the set , i.e. , , , .
(d) Let be a topological space, and let . The relative topology on (or the topology inherited from ) is the collection
of subsets of . It is clearly a topology on . The space is then called a subspace of .
(e) Let be a non-empty infinite set and be the family of open sets. Then is the so-called finite complement topology. First note, that an element , is of infinite cardinality. Think about what happens if we remove finitely many elements from an infinite set.
Apparently, since can be considered as finite. due to the definition of the set even though is not finite.
The union of arbitrary open as well as the intersection of a finitely many open sets are open again. The corresponding proof employs De Morgan’s Laws.
According to Proposition 1.1 – Fundamentals of Topology & Metric Spaces, the set of all open sets in a metric space complies with the definition of a topology.
Definition 1.2 (Induced & Equivalent Topology)
A topology induced by a metric space is defined as the set of all open sets in .
Two metrics and on the same basic set are called topologically equivalent if .
Let us consider illustrative examples.
Example 1.2 (Metrizable and Equivalent Topologies)
(a) If is a metric space and is the set of all open sets, then is a topology. The topology does not depend on the particular metric since the proof of the referred Proposition 1.1 can also be done using the term neighborhood instead of the distance function. Hence, any metric on equivalent to yields the same topology. Topological spaces of this kind are called metrizable.
with and . Instead of using the Euclidean metric, we could also employ the following distance functions:
All three induced topologies would be equivalent, i.e. since
for all . The corresponding unit open balls centered at are illustrated as follows.
An open ball with as shown in Fig. 1 is actually a set of points where each point has a distance of max. 1 to the origin . This, however, is nothing but the corresponding norm . For instance, the point is not an element of the unit ball induced by since . However, the same point is element of the unit balls induced by and .
From a topological point of view the shapes in Fig. 1 are all equivalent.
Definition 1.3 (Closed Sets)
A set is called closed in if is open in .
Example 1.3 (Closed Sets)
(a) Let b a metric space. Then is closed in if and only if is closed in .
(b) The sets are not only open but also closed for any topological space since and . The topological space is indiscrete if and only if these two sets are the only closed sets in .
(c) The topology is discrete if and only if every subset is closed. This can be seen by .
(d) The subset of is closed because its complement is open. Similarly, is closed, because its complement is open. The subsets of is neither open nor closed.
(e) In the finite complement topology on an infinite set , the closed sets consists of itself and all finite subsets of .
Proposition 1.1: (Characterization of Closed Sets)
(1) The set of all closed sets of complies with the following conditions:
(i) and .
(ii) implies .
(iii) implies .
(2) Let be a family of sets that complies with (i), (ii) and (iii) then there exists a topology , such that is the set of all closed sets in .
Proof. (1) This follows directly from the definitions and applying the rules , , as well as .
(2) The family of closed sets fully determines the topology on the same basic set since . Its existence follows from the fact that actually is a topology but this is clear given (1).
The family of closed sets of a topology could also be used to define a topological space, i.e. the set of all closed sets contains exactly the same information as the set of all open sets that actually define the topology.
Definition 1.4 (Interior & Closure)
Let be a topology. The interior of is defined as the union of all open sets contained in , i.e.
The closure of is defined as the intersection of all closed sets containing , i.e.
A neighborhood of a point in a topological space is any set containing as well as an open set of , i.e. .
A set is open in a Euclidean metric space if and only if for every an open ball exists such that . Let us therefore consider the situation in the real plane employing the topology induced by the Euclidean metric. The open balls and are both contained in and thus a part of the interior Int. Thereby, can be considered as neighborhoods of with . A is a neighborhood of .
The open ball is not fully contained in , which is why is not a neighborhood of and also not an element of the interior of .
The dual situation of the closure Cl can also be seen in the figure above by considering . An important type of points in such situations is the following.
Definition 1.2 (Boundary Point)
Let , and . Then, is called boundary point for if
for all . The set of all boundary points is denoted by .
Boundary points are just points on the boundary between the set and the surrounding basis set (i.e. of the metric space .
can be contained in or in . If all boundary points are outside of , i.e. if , then it is an open set.
Considering Example 1.2 (a), we could ask whether every topological space is metrizable?
The answer is no, and the root-cause is that topological spaces have different types of separation properties.
A metric enables us to separate points in a metric space since any two distinct points have a strictly positive distance. In general topological spaces, separating points from each other is more subtle.
Hausdorff spaces and the Hausdorff condition are named after Felix Hausdorff, one of the founders of topology. Let us first check out the formal definition.
Definition 2.1: (Hausdorff Space, Spaces)
A topological space is a Hausdorff or -space if, for any pair of distinct points , there are disjoint open sets with , and .
Every Euclidean space is Hausdorff since we can use the Euclidean metric to separate two distinct points. The following video outlines the Hausdorff condition and it provides a simple example of a Hausdorff space.
Example 2.1 (Metric Space is Hausdorff)
Let be a metric space, and let be such that . It follows that . Let and . Then, is Hausdorff since .
In a Hausdorff space, distinct points can be separated by open sets.
The situation in along with the topology implied by the Euclidean metric is illustrated in the graph above. For two distinct points , we take half (or less) the distance to define to come up with two distinct open balls, that can also be seen as disjoint neighborhoods.
Proposition 2.1: (Subset of -spaces)
Let be a topological Hausdorff space. Then, each subset of a Hausdorff space is Hausdorff.
Proof. Let be in . The space being Hausdorff, let and be the two open separating sets as required in Definition 1.1. Then as well as are open since the difference of two open sets is open. In addition, and .
– and – Spaces
The following separation properties are weaker than the Hausdorff (-) condition. This is also indicated by the index of the corresponding names of the separation axioms (from to ).
Definition 2.2. ( Space)
A topological space is called a – or Kolmogorov space if, for any with , there is an open set with and or and .
The most striking difference between a Hausdorff / – and -space is that only one open set , that contains only one of two distinct points, is required to fulfill the definition of a -space. Apparently, every -space is also a -space.
(a) Let be any set with at least two elements equipped with the so-called chaotic topology . Then, there is no open that separates the two distinct elements. Hence, this topology is not a -space and definitely also not a Hausdorff space. Hence, it is also not metrizable.
(b) Let be any set with the discrete topology . Then, separates the two elements and .
In a -space two open sets are required to separate two distinct points, however, the two sets don’t need to be disjoint.
Definition 2.3. ( Space)
A topological space is called a -space if, for any with , there are open sets with and and and .
The main difference between a – and a -space is that the two required open sets do not have to be disjoint. However, one open set only contains one of the two distinct points.
Hence, every -space is also a -space. Just take one of the two open sets of the -space and it fulfills all requirements of a -space.
Proposition 2.2: (Characterization of -spaces)
(a) Let be a topological space. Then, is a -space if and only if is a closed set for each ;
(b) Each Hausdorff space is a -space.
Proof. (a) Suppose is a -space, and let . For any with , there is an open subset of with , but . It follows that .
Conversely, suppose that all singleton subsets of are closed, and let be such that . Then, and fulfill the requirements of a -space.
(b) Let be a given point. By assumption, each belongs to an open set such that . Consequently, . Thus, is open, and is closed.
One of the key features of topological spaces is the generalization of the convergence concept.
A sequence in a (metric) space is a function that we also denote by , in particular, if we want to refer to the elements of the sequence. Given a sequence in a metric space, a sub-sequence is the restriction of to an infinite subset . If we exhibit as , then we write the subsequence .
We say that a sequence converges to if given , there exist such that for all , we have .
In other words, for all , we have as illustrated in Figure above. The finitely many elements of the sequence are, however, not contained in . We take this property to define what a convergent sequence in a topological space is.
Definition 3.1. (Convergent Sequence)
Let be a topological space. A sequence converges to if , the set is finite for any open set .
The point is then called the limit of the sequence and we denote it by or by .
Note that the set of is infinite for all open sets while (at the same time) the set is finite. Both sets/conditions matter in this situation as we will see further below!
Lemma 3.1. (Limit of a sequence is unique)
The limit of a convergent sequence in a Hausdorff space is unique.
Proof. Assume that this is not the case and as well as with holds true. A metric space is Hausdorff, that is, we find two disjoint open balls and . Given that and are the limit points almost all elements must lie in the disjoint balls, which contradicts the initial assumption of .
Lemma 1.1 is false in arbitrary topological spaces.
Every of a topological space is the limit of a certain sequence . Apparently, we could simply use the constant sequence or we could define for all , . This fact should be also considered in the following examples.
(a) Let be the discrete topology with . Further, let . Recall that in this topology every set is open by definition. Hence, also is an open set that must be contained in all other (open) supersets. Hence, the set has to be finite and has to be infinite.
(b) Let be the indiscrete topology with . Further, let . Since is the only set that contains , the set has to be finite for every sequence . Hence, every sequence converges to every point in .
Closely related to converging sequences and their limits are accumulation points.
Definition 3.2. (Accumulation Point)
An element of a sequence is called accumulation point (sometimes also cluster or limit point) if is infinite for every open set of .
The subtle but important difference between an accumulation point and a limit is that the complement set of can also be infinite. Let us consider a simple example.
Let us consider the sequence in the topology induced by the Euclidean space on the real line. There are two accumulation points but no limit of the sequence. Note that the sequence is alternating between and , such that and are both infinite but disjoint to each other. In addition, the set for an open set of are both infinite.
A sub-sequence of a convergent sequence converges to the same limit . This is evident since if the condition of a convergent sequence is fulfilled for all elements , of the series . Hence, the condition is also fulfilled for a subset that represents the sub-sequence.
Due to the fact that the finiteness of implies the infiniteness of in every limit is an accumulation point. The converse is not true as we can see in Example 3.2.
Theorem 3.1 (Convergence in Topological Spaces)
Let be a topological space.
(i) Every limit of a convergent sequence is also the limit of any sub-sequence.
(ii) Every accumulation point of any sub-sequence is also an accumulation point of .
Proof. (i) If then is finite for all open of . In particular, this holds true for any sub-sequence and thus .
(ii) Let be an accumulation point of the sub-sequence, i.e. the set is infinite for every open set of . Since the sub-sequence is only a subset of the element of the sequence, the assertion follows directly.
Completely Regular Spaces
Our next separation axiom has a somewhat different flavor; it is not defined in terms of topology, but via continuous functions.
Definition 4.1: (Completely Regular Space)
Let be a -space. Then is called completely regular if and only if, for every point and every closed set , , there is a function with , , and .
A completely regular space posses a family of corresponding functions with and for all closed . Let us consider an example.
Example 4.1: (Completly Regular Metric Space)
Let be a metric space, let , and let be closed such that . To avoid triviality, suppose . Then, define
Definition 4.2: (Locally Compact)
A topological space is locally compact if every point has a neighborhood whose closure is compact.
Example 4.2: (Locally Compact )
The topology where are the open sets of the Euclidean metric is locally compact.
According to the Heine-Borel Theorem, is closed and bounded if and only if is compact. Every point posses infinitely many open sets , and corresponding closed balls . The closed balls are bounded by definition (i.e. and they are also closed since its complement is open. Refer to Example 1.2 as well as Example 1.3 (d) in this post for further details. Thus, the set is compact and locally compact.
The compactness (i.e. boundedness and closedness) is needed in combination with continuous functions as we will see further below.
A subset of a locally compact space is bounded if there exists a compact set such that .
For any topological space , we denote by the class of all real-valued continuous functions such that for all .
If is a compact set and and are open sets such that , then there exist compact sets and such that , , and .
Proof. Since and are disjoint compact sets, there exist two disjoint open sets and , such that and . We write and . It is easy to verify that , , and that and are compact. Since , we have .
Theorem 1.1 tells us that if we have a two-element open cover of a compact set then we can find a two-element closed cover of .
Another important theorem is the following.
If is a compact set and is a closed disjoint set, i.e. . Then there exists a function in such that for and for .
Proof. Since is completely