Integridad

Knowing about affine spaces, it is a must to bear in mind a few topological notions and defini-tions associated with subsets of affine spaces. Let us start by the notion of a ball.

Definition 1.1.25. Given a point ψ in an normed affine space SV and a positive real number

 > 0, the subsets

S_ball^o = {φ|φ ∈ SV, kφ − ψk < }, (1.22) S_ball^c = {φ|φ ∈ SV, kφ − ψk 6 }, (1.23) are called open and closed balls in SV, respectively, centred at ψ with radius .

Whenever we speak of a ball S_ball without mentioning it being open or closed, we assume any of them. Then, we can define a neighbourhood as follows.

Definition 1.1.26. Given a point ψ in an affine space SV, the subset S ⊆ SV is called a neighbourhood of ψ if and only if there exists a ball S_ball(ψ) centred at ψ such that S_ball(ψ) ⊂ S.

A neighbourhood S of a point ψ in SV has the simple geometrical meaning that sufficiently close to ψ in SV, the subset S contains all points from the underlying space SV. Using the notion of balls in inner-product affine spaces, we can restate the idea of convergence of sequence of points in Definition 1.1.4 and Eq. (1.6) from a topological perspective as follows.

Definition 1.1.27. A sequence of points S_seq = (ψ_i)_i∈N in a normed affine space SV is conver-gent to the limit ψ if and only if for every  > 0 there exists a subsequence Sseq;N = (ψi)i>N ⊆ S_seq for some integer N ∈ N that belongs to the ball centred at ψ with radius . Equivalently, every neighbourhood of ψ on S_V intersects with S_seq (See Fig. 1.3).

Figure 1.3: Geometrical representation of a limit point. The point ψ is the limit point of the sequence S_seq = (ψ_i)_i∈N if for every ball of radios  centred at ψ there exists a subsequence of Sseq that belongs to that ball.

Moreover, subsets of affine spaces can be bounded to a finite region of the space. This can be rigorously defined as below.

Definition 1.1.28. A subset S of the affine space SV is called bounded if and only if there exists a ball Sball such that S is confined within that ball, i.e. S ⊂ Sball.

The boundedness on a metric space simply means that every two points of the set are within a finite distance from each other. We have now the ingredients to define two fundamental properties of subsets of affine spaces, namely openness and closedness.

Definition 1.1.29. A subset S of the affine space SV is called open if and only if it is a neighbourhood of each point ψ ∈ S.

In other words, for every point ψ in an open subset S there exists a ball centred at ψ such that the ball is fully contained in the set S.

Definition 1.1.30. A subset S of the affine space SV is called closed on SV if and only if its complement SV/S is open.

Proposition 1.1.6. A half-space as given by Definition 1.1.24 is closed.

It is also useful to define open half-spaces as follows.

Definition 1.1.31. Given a hyperplane S_hp;c^φ with the normal vector φ − φ0, it splits the whole affine space S_V into two open half-spaces given by

S_+;c^o = {ψ|ψ ∈ SV, x_i ∈ K, ψ =

It is interesting to note that although the notion of closedness for a set S is defined via the openness of the complementary set to S, closedness and openness are not exclusive properties.

That is, it is possible for a set S to be neither open nor closed, as is the interval (0, 1] o the real line R. Also, it is possible for S to be both open and closed (sometimes called clopen), as are

the empty set and the entire affine space SV. More importantly, closedness (and also openness) of a subset S depends strongly on the topology of the container space S_V.

Lemma 1.1.5. A subset S of an inner-product affine space S_V is closed on S_V if and only if the convergence point ψ of every sequence on S which is convergent on S_V is a member of S.

Lemma 1.1.5 highlights the role of the container topology and the particular convergence cri-terion exploited. For instance, while the interval (−1, 1) is open on the real line R, it is closed when the space containing this subset is the interval (−1, 1) itself, because its complement with respect to itself is the empty set which is open². In the language of Lemma 1.1.5, it is trivial that the subset (−1, 1) contains the convergence point of any sequence on (−1, 1) that is convergent on (−1, 1). Two useful properties of open and closed sets is given by the following proposition.

Proposition 1.1.7.

i. Any union of open sets is open

ii. A finite intersection of open sets is open.

iii. A finite union of closed sets is closed.

iv. Any intersection of closed sets is closed.

We now turn into the notion of completeness for subsets of affine spaces, similar to Defini-tion 1.1.18.

Definition 1.1.32. A subset S of an affine space SV is called complete if and only if every Cauchy sequence of points in S converges to a point ψ within S.

Thus, by comparing Definition 1.1.32with Lemma 1.1.5we see that completeness is a stronger notion than closedness, as it is independent of the underlying space of a subset. Reconsidering our example above, the interval (−1, 1) is not complete, independent of being considered on itself or R, because the sequence 1 + 1/n which is Cauchy with respect to the absolute value norm does not converge to a point within this interval. There is indeed a connection between closed and complete subsets of affine spaces as given in Lemma 1.1.5 and Definition 1.1.32, respectively, via the following theorem.

Theorem 1.1.2. [2] If an affine subset S of an affine space S_V is complete, then it is closed on SV. The converse is true only if SV is complete.

We have seen that the convergence point of a sequence plays a key role in topological properties of affine sets. It is thus reasonable to look at this particular point from topological perspective, which will be useful later on. Firstly, we define the convergence points in a way independent of the sequence converging to them.

2In fact, both the empty set and the (−1, 1) interval on itself are clopen sets.

Definition 1.1.33. A point ψ in an affine space SV is called a limit (or accumulation) point of a subset S ⊆ S_V if and only if every neighbourhood of ψ in S_V contains at least one point from S not equal to ψ itself. Equivalently, every ball S_ball(ψ) centred at ψ intersects with S in at least one point not equal to ψ itself.

Note that, a limit point of a set S may or may not be in S. Interestingly, it is possible to define the opposite of a limiting point, that is, an isolated point.

Definition 1.1.34. A point ψ from a subset S of an affine space SV is called an isolated point of S if and only if there is a neighbourhood of ψ in SV that contains no point from S except ψ itself. Equivalently, there exists a ball Sball(ψ) centred at ψ such that any point φ ∈ Sball and φ 6= ψ satisfies φ /∈ S.

Using Definition.1.1.27of a limit point, we infer that the isolated point is not the convergence point of any sequence in S. Conversely, if ψ is not the accumulation point of any sequence³ on S, then it is an isolated point of S. It also becomes clear that, in an inner-product space, for every limit point ψ it is always possible to construct a sequence on S that converges to ψ. Now, we use Definition1.1.27in combination with Lemma1.1.5and employ the above considerations to state the following equivalent definition of closedness of affine subsets.

Lemma 1.1.6. A subset S of an affine space SV is closed on SV if and only if for every ψ ∈ S every neighbourhood of ψ on SV intersects with S.

As a result, every point of a closed set S is either a limit point or an isolated point of S [2].

Corollary 1.1.2. Every point of a closed set S is either a limit point or an isolated point of S.

We have seen that open and closed subsets are two important classes of points in an affine space. There are qualities of subsets that particularly separate open and closed sets. Below we identify such qualities.

Definition 1.1.35. A point ψ within a subset S of an affine space SV is called an interior point of S if and only if S is a neighbourhood of ψ.

Definition 1.1.36. A subset intS of another subset S of an affine space SV is called the interior of the subset S ⊆ SV if and only if it is the collection of all interior points of S. Equivalently, intS is the largest open set contained in S.

The interior points and interior subset of a set are both abstract definitions of what we intu-itively understand as “inside”. However, it is very important to note that, like many other topo-logical properties, the definition above depends on the container affine space SV. For instance, while a two-dimensional circle S = {(x, y)|x, y ∈ R, x²+ y² 6 1} defined in the two-dimensional Euclidean vector space R² has the interior points satisfying x²+ y² < 1, it has no interior points

3Except the sequence which is constant except at finitely many points.

when defined within the 3-dimensional Euclidean space R³ as S = {(x, y, 0)|x, y ∈ R, x²+ y² 6 1}. The reason for the latter is that, considering Definitions 1.1.25,1.1.26, and1.1.35, any ball around any point of S in R³ contains points that are not within S.

A similar approach exists for the notion “outside”.

Definition 1.1.37. A point ψ of an affine space SV is called an exterior point of S if and only if it is an interior point of the complement SV/S of S.

Definition 1.1.38. A subset extS of an affine space S_V is called the exterior of the subset S ⊆ S_V if and only if it is the collection of all exterior points of S. Equivalently, extS is the largest open set contained in SV/S.

Besides interior and exterior sets to a given subset of an affine space, we also have the similar statement to the Proposition 1.1.3.

Definition 1.1.39. A subset ¯S of an affine space S_V is called the closure of the subset S on SV if and only if it is the smallest closed subset of SV containing S.

As we stated in Definitions 1.1.36 and 1.1.38, interior and exterior sets are open sets. Thus intuitively we can define the boundary of a subset as follows.

Definition 1.1.40. A subset ∂S of an affine space S_V is called the boundary of the subset S on SV if and only if it holds true that ∂S = ¯S/intS.

Thus, we see that every subset of an affine space splits the space into three regions, interior, exterior, and boundary points. Interestingly, there is an intuitive proposition regarding closed subsets and boundary subsets as follows.

Proposition 1.1.8. [2] A subset S of an affine space SV is closed if and only if it contains its boundary, that is, ∂S ⊂ S.

As a result of Proposition1.1.8 and Definition1.1.40we see that when S is closed, i.e., ¯S = S, we have

intS = S/∂S, or equivalently, intS ∪ ∂S = S. (1.25) At this point, it is appropriate to state the following connection between limit points and closedness of sets.

Theorem 1.1.3. [2] Suppose that S_lim is the set of all limit points of S. Then, the closure ¯S of S is given by

S = S ∪ S¯ _lim. (1.26)

Corollary 1.1.3. A subset S of an affine space S_V is closed if and only if it contains its limit points, that is, S_lim⊂ S.

The similarity between Eqs. (1.25) and (1.26) motivates the question that “what is the difference between the set of all limit points and the boundary?” To answer that, we first notice the fact that, by Definitions 1.1.25 and 1.1.26 of a ball and a neighbourhood, an isolated point is not an interior point, as given by Definitions 1.1.34 and 1.1.35, respectively. Employing this in Definition 1.1.40 of the boundary set, we conclude that isolated points of the closure ¯S of a subset S belong to the boundary of S. On the other hand, however, it is clear that by definition isolated points are not limit points. Consequently, isolated points of the closure ¯S are examples of the points which are within the boundary of S, but not within S_lim. Hence, we arrive at the following.

Corollary 1.1.4. Not every point within the boundary is a limit point.

The converse also is not true, because by comparing Definitions 1.1.33and 1.1.35, we have the following result⁴.

Corollary 1.1.5. Any interior point of a subset S of an inner-product affine space is a limit point of S.

Hence, by the subtraction operation in Definition 1.1.40, the limit points interior to the set are not within the boundary.

Corollary 1.1.6. Not every limit point is within the boundary.

It is also useful to note that, using Definition 1.1.37, the following statement holds true.

Corollary 1.1.7. Any exterior point of a subset S of an inner-product affine space is not a limit point of S.

To give an explicit example for some of the above definitions, consider the interval (0, 1] of the real line R. Trivially, its interior set is (0, 1), the set (−∞, 0) ∪ (1, +∞) is its exterior, and {0, 1} forms its boundary. Moreover, on top of the interior points, the point {0} is also a limit point for this subset. We can also verify that (0, 1) ∪ {0, 1} = (0, 1] ∪ {0} = ¯S. At this point, it is important to note that the concepts interior, exterior, and boundary depend on the container space SV, in the same way as the more primitive notions of openness and closedness depend on it.

By looking at Definition 1.1.33 we see that if a point ψ ∈ SV is a limit point of the subset S ⊆ SV, then it is possible to approximate ψ from the larger set SV to arbitrary precision with an element of the special subset S which is arbitrary close to ψ. In fact, the concept of limit points is the abstract way of thinking about such approximation procedures. Consider now an inner-product affine space SV and a subset S such that every element of SV can be approximated to arbitrary precision with an element of S. This is equivalent to say that any point of SV is a limit point for S. For this to be true, we use Corollary1.1.7to conclude S must

4Note that, in a more general topological sense, this statement is not true because in that case interior points are defined using open sets rather than balls. This makes it possible to have interior points which are not limit points as in the example of discrete topological spaces.

Figure 1.4: Geometrical representation of a cover. A set is covered by a finite collection of balls if it is a subset of their union. A set is compact if for any of its covers there exists a finite subcover.

not have any exterior points, i.e. extS = ∅. Due to the fact that the whole space is divided into three subsets, that is SV = intS ∪ extS ∪ ∂S, we arrive at

SV = intS ∪ ∂S = ¯S = S ∪ S_lim. (1.27) This read as every point of SV can be approximated to arbitrary precision using elements of S if and only if the closure of S is the whole space SV. In the technical terms, such a subset S is called a dense subset in SV.

Definition 1.1.41. A subset S of an inner product affine space S_V is called dense in S_V if and only if its closure ¯S is the whole space, i.e. S_V = ¯S.

It is also interesting to consider the case in which no point of SV can be approximated by points within S, that is, S has no limit points. The latter, has two consequences. First, by using Theorem1.1.3, S is closed, i.e. S = ¯S. Second, by Corollary 1.1.5, the interior of S must be empty, i.e., intS = ∅. By combining the two we find that int ¯S = ∅. Hence, we define the following.

Definition 1.1.42. A subset S of an inner product affine space SV is called nowhere dense in SV if and only if its closure ¯S has no interior points, i.e. int ¯S = ∅.

Finally, we may use the term “compact sets”.

Definition 1.1.43. Given a set S, it is said to be covered by a (possibly infinite) collection of balls S_cover = {S_ball(ψ_i)} centred at points ψ_i ∈ S if and only if

[

i

Sball(ψi) ⊇ S. (1.28)

Definition 1.1.44. A set S is said to be compact if and only if for every cover Scover = {S_ball(ψ_i)} there exists a finite subcover Scover⁽ⁿ⁾ = {S_ball(ψ_i)}ⁿ_i=1 ⊂ S_cover such that

n

[

i=1

S_ball(ψ_i) ⊇ S. (1.29)

The notion of compactness can be understood topologically in two levels. First, it puts a divisibility property on sets, meaning that, no matter how one divides a set into possibly infinite number of parts, always a finite number of those parts are sufficient to reconstruct the set; see Fig. 1.4. The following thus immediately results.

Corollary 1.1.8.

i. Every finite set is compact, because it cannot be divided otherwise into a finite number of parts.

ii. A finite union of compact sets is compact.

iii. Any intersection of compact sets is compact.

On a second level, through this division property, compactness induces a topological meaning for finiteness. Topological objects can be deformed continuously, e.g., by stretching. Therefore, concepts like area or volume do not capture the essence of finiteness in topology and thus, one requires a notion which does so. Now, note the following proposition.

Proposition 1.1.9. [2] If f : S_V → S_V is a continuous map and S ⊂ S_V is compact, then f (S) is also compact.

In particular, suppose that S_cover and Scover⁽ⁿ⁾ ⊂ S_cover are a cover and a finite subcover for S, respectively. It can be readily shown that f (S_cover) and f (Scover⁽ⁿ⁾ ) are the corresponding cover and subcover of f (S). The important point is that, in a sense, the number n of required subsets to resemble S does not increase under the continuous map f and thus, the finite characteristic of S is preserved by f . Loosely speaking, one can think of n as a replacement for quantities like area and volume, characterizing the finiteness of S irrespective of continuous topological transformations⁵.

As discussed above, it is expected from compactness to be in close connection with closedness and boundedness.

Proposition 1.1.10. [2]

i. A closed subset of a compact set is compact.

ii. Compact sets are closed and bounded.

iii. In a metric space, a subset S is compact if and only if every sequence in S has a subsequence that converges in S.

5Note that by no means n is not unique, as it depends on which subcover has been chosen.

In particular, we may state a result due to Heine and Borel for finite dimensional spaces.

Theorem 1.1.4. (Heine-Borel Theorem.) [3] Any subset of a finite dimensional normed affine space is compact if and only if it is closed and bounded.

For instance, the interval [0, 1] ⊂ R is compact. Note that in infinite dimensions this theorem does not necessarily holds true.