9.1 Topology

Having completed an overview of linear and multilinker algebra, we now turn our attention to topology. This is an extensive area of mathematics, and of necessity the coverage presented here is highly selective. To the uninitiated, topology can be dauntingly abstract. The best example of a topology (and the motivation for much of what follows) is the Euclidean topology on ℝ ^m , covered briefly in Section 9.4 and preceded by preparatory material in Section 9.2 and Section 9.3. Readers new to topology might find it helpful to peruse these sections early on to get a glimpse of where the discussion below is heading.

A topology on a set X is a collection ? of subsets of X such that:

• [T1] ∅ and X are in ?.
• [T2] The union of any sub collection of elements of ? is in ?.
• [T3] The intersection of any finite sub collection of elements of ? is in ?.

The pair (X, ?) is referred as a topological space and each element of ? is said to be an open set in ? or simply open in ?. Each element of X is called a point in X. Any open set in ? containing a given point x in X is said to be a neighborhood of x in X . We say that a subset K of X is a closed set in ? or simply closed in ? if is an open set in X. It is often convenient to adopt the shorthand of referring to X as a topological space, with ? understood from the context. Accordingly, if U is an open set in ? and K is a closed set in ?, we say that U is an open set in X or simply open in X , and that K is a closed set in X or simply closed in X .

Let X be a topological space, and let S be a subset of X. The interior of S in X is denoted by int _X (S) and defined to be the union of all open sets in X contained in S. The exterior of S in X is denoted by ext _X (S) and defined to be the union of all open sets in X contained in . The boundary of S in X is denoted by bd _X (S) and defined to be the set of all points in X that are neither in int _X (S) nor in ext _X (S). Thus, X is the disjoint union

The closure of S in X is denoted by cl _X (S) and defined to be the intersection of all closed sets in X containing S.

Here are some of the basic facts about interiors, exteriors, boundaries, and closures.

Rather than having to deal with all open sets in X, it is often convenient to work with a smaller collection that contains the essential information on “openness”. A basis for X is a collection ℬ of subsets of X, each of which is called a basis element, such that:

• [B1] X =  ∪ B ∈ ℬB .
• [B2] If B ₁, B ₂ are basis elements in ℬ and x is a point in B₁ ∩ B₂, then there is a basis element B in B containing x such that B ⊆ B ₁ ∩ B ₂.

In the notation of Theorem 9.1.4, we say that (S, ? _s ) is a topological subspace of (X, ?) or simply that S is a topological subspace of X.

Throughout, any subset of a topological space is viewed as a topological subspace.

Let X and Y be topological spaces, let F : X → Y be a map, and let x be a point in X. We say that F is continuous at x if for every neighborhood V of F(X) in Y, there is a neighborhood U of x in X (possibly depending on V) such that F(U) ⊆ V . Equivalently, F is continuous at x if for every neighborhood V of F(x) in Y, F ⁻¹(V) contains a neighborhood of x in X. We say that F is continuous on X or simply continuous if it is continuous at every x in X.

Let X and Y be topological spaces, let F : X → Y be a map, and let S be a subset (topological subspace) of X. We say that F is continuous on S if the restriction map F| _S : S → Y is continuous on S.

Let X and Y be topological spaces, and let F : X → Y be a continuous map. We say that F is a homeomorphism, and that X and Y are homeomorphism, if F is objective and F ^–1 is continuous. The next result shows that a homeomorphism is a map that preserves topological structure (in much the same way that a linear isomorphism preserves linear structure).

We say that a topological space X is disconnected if there are disjoint nonempty open sets U and V in X such that X = U ∪ V. In that case, U and V are said to disconnect X. If X is not disconnected, we say it is connected. A subset S of X is said to be disconnected (connected) in X or simply disconnected (connected) if it is disconnected (connected) as a topological subspace of X. We say that a subset C of X is a connected component of X if it is a connected set in X that is maximal, in the sense that C is not properly contained in any other connected set in X.

The next result gives equivalent conditions for a subset of a topological space to be disconnected in the topological space.

Let X be a topological space, and let 풰 = {U _α : α ∈ A} be a collection of open sets in X. We say that 풰 is an open cover of X if X = U _α∈A U _α. A sub collection of 풰 that is also an open cover of X is said to be a subcover of 풰. A subcover that is a finite set is called a finite subcover. We say that X is a compact topological space if every open cover of X has a finite subcover. A subset S of X is said to be a compact set in X or simply compact in X if it is compact as a topological subspace.

Let X be a topological space, and let f : X → ℝ be a function. We say that f is bounded on X or simply bounded if there is a real number c > 0 such that |f (x)| < c for all points x in X. Let S be a nonempty subset of ℝ. The element s ₁ in S is said to be the smallest element in S if s ₁ ≤ s for all s ₂ in S. Similarly, the element s ₂ in S is said to be the largest element in S if s < s ₂ for all s in S. If S has a smallest element s ₁ and a largest element s ₂, then s ₁ ≤ s ≤ s ₂ for all s in S. Consider the collection 풰 = {(-∞, s) ∩ S : s ∈ S} of open sets in S. If S has a largest element s ₂, then s ₂ is not in ∪_s∈S (‐∞, s), so 풰 is not an open cover of S in ℝ. On the other hand, if S does not have a largest element, then 풰 is an open cover of S in ℝ.

Let X be a topological space, and let f : X → ℝ be a function. The support of f is denoted by supp(f) and defined to be the closure in X of the set of points at which f is no vanishing:

Thus, supp(f) is the smallest closed set in X containing those points at which f is no vanishing. We say that f has compact support if supp(f) is compact in X. Given a subset S of X, we say that f has support in S if supp(f) ⊆ S.

9.2 Metric Spaces

Let X be a nonempty set X. A function

is said to be a distance function on X if for all x, y, z in X:

A metric space is a pair (X, d) consisting of a nonempty set X and a distance function d on X. For a given point x in X and real number r > 0, the open ball of radius r centered at x is defined by

and the closed ball of radius r centered at x by

The next result justifies our use of the terms “open” and “closed” to describe balls in a metric space.

The next result expresses continuity in metric spaces in terms familiar from the differential calculus of one real variable.

9.3 Normed Vector Spaces

Let V be a vector space. A function

is said to be a norm on V if for all vectors v, w in V and all real numbers c:

A normed vector space is a pair (V, ||·||) consisting of a vector space V and a norm ||·|| on V.

Theorem 9.3.1 justifies our use of the term “norm” in Section 4.6 (and to a lesser extent its use in Section 4.1).

9.4 Euclidean Topology on ℝ^m

Let us put the results of Sections 9.1–9.3 to work constructing a series of “spaces” starting with the inner product space , as defined in Example 4.6.1. We first apply Theorem 9.3.1 to (ℝ ^m , e) and obtain a normed vector space (ℝ ^m , ||·||), where

is the norm introduced in Example 4.6.1 and defined by

We next apply Theorem 9.3.2 to (ℝ ^m , ||·||) and obtain a metric space (ℝ ^m ,∂ ), where

is the distance function defined by

Lastly, we apply Theorem 9.2.2 to (ℝ ^m , ∂) and obtain a topological space (ℝ ^m ,∂).

Recall from Example 4.6.1 that e is called the Euclidean inner product. In a corresponding fashion, we refer to ||·|| as the Euclidean norm on ℝ ^m , to d as the Euclidean distance function on ℝ ^m , and to T as the Euclidean topology on ℝ ^m . With the above constructions, there is now a direct path from (ℝ ^m , e) to (ℝ ^m , ||·||) to (ℝ ^m , ∂) to (ℝ ^m , ). We generally simplify notation by using ℝ ^m to denoted any of the latter three spaces, allowing the context to make it clear whether ℝ ^m is being thought of as a normed vector space, a metric space, or a topological space. On occasion, we also adopt ℝ ^m as notation for (ℝ ^m , e).

Let us discuss a few examples based on the Euclidean topologies on ℝ and ℝ² to illustrate some of the material in Sections 9.1–9.3.

We first consider ℝ. An open (closed) “ball” in ℝ is nothing other than a finite open (closed) interval. By Theorem 9.1.3 and Theorem 9.2.2, the collection of open intervals in ℝ is a basis for the Euclidean topology of ℝ. An open set in ℝ is obtained by forming an arbitrary union of open intervals, and a closed set in ℝ results from taking the complement in ℝ of such an open set. For example, (0,1) and (0, +∞) are open in ℝ, whereas [0,1] and {0} are closed in ℝ. Working through the definitions, we find that (0,1) is the interior in ℝ of both (0,1) and [0,1], and that {0,1} is the boundary in ℝ of both (0,1) and [0,1]. Consider the functions f ₁, f ₂, f ₃ : ℝ → ℝ given by f ₁(x) = x ³, f ₂(x) = |x|, and

Then f ₁ and f ₂ are continuous, but f ₃ is not. In fact, f ₁ is a homeomorphism.

We now turn our attention to ℝ². The open “ball” (actually “disk”) in ℝ² of radius ℝ centered at (x ₀,y ₀) is

For example,

is the unit open disk in ℝ² (centered at the origin). By Theorem 9.1.3 and Theorem 9.2.2, the collection of open disks in ℝ² is a basis for the Euclidean topology of ℝ². An open set in ℝ² is obtained by forming an arbitrary union of open disks, and a closed set in ℝ² results from taking the complement in ℝ² of such an open set. For example,

is open in ℝ², whereas [0,1] and

are closed in ℝ².

Suppose D, as given above, has the subspace topology induced by ℝ². Consider the map F : D → ℝ² (between topological spaces) defined by

It can be shown that F is a homeomorphism, with inverse F ^–1 : ℝ² → D given by

Thus, D is homeomorphism to all of ℝ². Now consider the subsets {(x, 0) : x ∈ ℝ} and {(x, |x|) : x ∈ ℝ} of ℝ², and suppose each has the subspace topology induced by ℝ² . It can be shown that the map

(between topological spaces) given by G(x, 0) = (x, |x|) is a homeomorphism. Thus, a “straight” line is homeomorphism to a line with a “corner”.

The next result shows that in the Euclidean setting a homeomorphism has something to say about “dimension”.

We close this section with a few remarks on compactness, perhaps the least intuitive of the concepts introduced in Section 9.1. As it stands, determining whether a subset of a topological space is compact is a seemingly complicated task. However, in the Euclidean case, matters are more straightforward. A subset S of ℝ ^m is said to be bounded if it is contained in some open ball of finite radius.

Thus, [0,1] × [0,1] is compact in ℝ², but (0,1) × (0,1) and ℝ × {0} are not. It should be emphasized that the preceding theorem rests on unique features of ℝ ^m and does not extend to an arbitrary metric space.