Topological Spaces
The $n$-dimensional Euclidean space, \(\mathbb{R}^n\), is the richest space that we know of. Notions of distance, norm, inner product etc. are understood in
\(\mathbb{R}^n\). Moreover, limits, continuity and differentiability are well defined on it. We would like to see how these concepts (or
more abstract versions of them) could be generalized to other spaces
that are not as rich as Euclidean spaces. Topological spaces are the most
abstract kind of such spaces. We shall begin the discussion on Riemannian manifolds by defining topological spaces.
Definition 1.1: Topological space
A set \(X\) together with a collections of its subsets \(T\), usually denoted by $(X, T)$, is known as a topological space if the following axioms are satisfied.
- \(X\) and \(\phi\) are included in \(T\).
- Any union of sets in \(T\) is included in \(T\).
- Any finite intersection of sets in \(T\) is included in \(T\).
In such cases, sets in \(T\) are named as open sets and $T$ itself is referred to as a topology of X. An open neighborhood of a point \(x\) \(\in\) \(X\) is any open set containing \(x\). In this series of articles we use the terms neighborhood and open neighborhood interchangeably to mean the same.
Example
$(X, T)$ with \(X = \{1, 2, 3\}, T = \{\{1,2,3\}, \phi, \{2, 3\}, \{2\}, \{3\}\}\) is a topological space since all the three axioms are satisfied. Some tests are shown below.
\(\{2\} \in T\) and \(\{2, 3\} \in T\),
- \(\{2\} \cup \{2, 3\} = \{2, 3\}\) is also in \(T\) (Axiom 2)
- \(\{2\} \cap \{2, 3\} = \{2\}\) is also in \(T\) (Axiom 3)
Similarly, the reader can verify the same for other unions and finite intersections of sets in \(T\). In this example \(\{1, 2, 3\}, \{2, 3\}\) and \(\{2\}\) are open neighborhoods of 2 \(\in\) \(X\).
Non-Example
$(X, T)$ with \(X = \{1, 2, 3\}, T = \{\{1,2,3\}, \phi, \{1, 2\}, \{2, 3\}\}\)
is not a topological space. Because, for \(\{1, 2\}, \{2, 3\} \in T\), their intersection, \(\{1, 2\} \cap \{2, 3\} = \{2\}\) is not in \(T\).
From now on we may refer to a set \(X\) as a topological space without explicit reference to its topology \(T\). In such references it is implied that a valid topology is defined on \(X\).
Euclidean Topology and Metric Topology
In order to understand the rather abstract notion of a topological space presented in the previous section, let us now consider a particular concrete example of a topological space. We pick the $n$-dimensional Euclidean space, $\mathbb{R}^n$. As understood from the previous section, in order to define a topological space, we need to identify a set ($X$) and a collection of subsets ($T$) that are closed under union and finite intersection. 'Open sets' is a name given to members of $T$. In the present example, $X$ is the infinite set that consists of all $n$-dimensional vectors in $\mathbb{R}^n$. What is $T$ then?
In order to define $T$, we first define open sets on $\mathbb{R}^n$ such that the axioms in Definition 1.1 are satisfied. Then, all $n$-dimensional vectors together with the collection of such defined open sets will form a topological space.
We now proceed to defining an open set in $\mathbb{R^n}$. We start with the definition of an open ball.
In order to define $T$, we first define open sets on $\mathbb{R}^n$ such that the axioms in Definition 1.1 are satisfied. Then, all $n$-dimensional vectors together with the collection of such defined open sets will form a topological space.
We now proceed to defining an open set in $\mathbb{R^n}$. We start with the definition of an open ball.
Definition 1.2: Open ball (Euclidean spaces)
For any \(x_0 \in \mathbb{R}^n\) and $\varepsilon > 0$, the open ball of radius $\varepsilon$ around $x_0$ is the set \(B_{\varepsilon}(x_0) = \{x \in \mathbb{R}^n \mid \|x - x_0\| < \varepsilon \}\).
An open ball around a point \(x_0 \in \mathbb{R}^2 \) is graphically shown in Figure 1. Radius of the ball, $\varepsilon$, can be arbitrarily small. The intuition is, an open ball contains all points that are sufficiently close to the given point. How do we define 'closeness'? Well, we use the concept of distance between two points in \(\mathbb{R}^n\).
Figure 1: An open ball in \(\mathbb{R}^2\). Note that the circumference of the circle is not included in the 'open' ball.
It's worth noting that this definition of an open ball depends only on the notion of distance between two points i.e., \(\| x - x_0 \| \). In Euclidean spaces, we measure the distance between any two points by taking the vector norm of the difference vector. This is known as Euclidean distance or Euclidean metric. It is not necessary to have a Euclidean space (or any normed vector space for that matter) to measure the distance between two points. In fact, the notion of distance between any two points is present in a more general metric space. While discussing about $\mathbb{R}^n$ we would like to extend the concepts to this more general structure.
Simply put, a metric space is a set with the notion of distance between two points. A metric space is usually denoted by $(M, d)$ where $M$ is the set and $d$ is the distance function or the metric. Any $n$-dimensional Euclidean space, along with the Euclidean distance, forms a metric space.
The following definition of an open ball in a metric space is a direct generalization of the above definition for Euclidean spaces.
The following definition of an open ball in a metric space is a direct generalization of the above definition for Euclidean spaces.
Definition 1.3: Open ball (Metric space)
Let $(M, d)$ be a metric space. For any \(x_0 \in M\) and $\varepsilon > 0$, the open ball of radius $\varepsilon$ around $x_0$ is the set \(B_{\varepsilon}(x_0) = \{x \in M \mid d(x, x_0) < \varepsilon \}\).
We next present the definition of an open set for a general metric space. This also includes the definition of an open set in Euclidean spaces since they are metric spaces.
Definition 1.4: Open set (Metric space)
Let $(M, d)$ be a metric space. A subset $A \subseteq M$ is said to be an open set if it contains an open ball around each of its points.
or equivalently,
An open set is a subset of $M$ that can be realized as a union of open balls in $M$.
Figure 2 visualizes an open set in $\mathbb{R}^2$. Note that an open ball itself is an open set according to the above definition. Also recall that an open neighborhood of a point is defined as an open set containing the point. Hence, whenever we refer to an open neighborhood of some point $x_0$, it is sufficient to visualize an open ball around $x_0$ with an arbitrarily small radius.
or equivalently,
An open set is a subset of $M$ that can be realized as a union of open balls in $M$.
Figure 2. An open set in $\mathbb{R}^2$. An open ball itself is an open set.
Figure 2 visualizes an open set in $\mathbb{R}^2$. Note that an open ball itself is an open set according to the above definition. Also recall that an open neighborhood of a point is defined as an open set containing the point. Hence, whenever we refer to an open neighborhood of some point $x_0$, it is sufficient to visualize an open ball around $x_0$ with an arbitrarily small radius.
Reader is encouraged verify that following Definition 1.4, open sets of a metric space are closed under union and finite intersection and therefore satisfy axioms in Definition 1.1. Therefore, a metric space (and hence the $n$-dimensional Euclidean space) together with the collection of open sets defined as in Definition 1.4 is a topological space. In the Euclidean case this topology is referred as the Euclidean topology and in the more general case of a metric space it is referred to as the metric topology.
Continuity
We will now discuss the topological definition of continuity of a function, perhaps the most important concept defined on topological spaces. We start from the usual \(\varepsilon\)-\(\delta\) definition of continuity of a function at a point in $\mathbb{R}^n$. The intuition is, a function $f$ is continuous at $x_0$ if it is possible to make \(f(x)\) arbitrarily close to \(f(x_0)\) by choosing \(x\) values sufficiently close to \(x_0\). It is formalized as follows:
Definition 1.5: Continuity at a point (Euclidean space)
A function $f: \mathbb{R}^n \to \mathbb{R}^m$ is said to be continuous at \(x_0 \in \mathbb{R}^n\) if \(\forall \varepsilon > 0, \exists \delta > 0\) such that, \(\|x - x_0 \| < \delta \Rightarrow \|f(x) - f(x_0)\| < \varepsilon \).
Following Definition 1.2, this is equivalent to stating that it is possible to confine $f(x)$ to an arbitrarily small open ball around $f(x_0)$ by confining $x_0$ to a sufficiently small open ball around $x_0$.
Definition 1.5 could easily be generalized to a metric space by replacing difference vector norms with the distance metric. Further, we can convert this definition to an equivalent involving open neighborhoods, using the definition of an open neighborhood in Euclidean spaces.
Definition 1.5 could easily be generalized to a metric space by replacing difference vector norms with the distance metric. Further, we can convert this definition to an equivalent involving open neighborhoods, using the definition of an open neighborhood in Euclidean spaces.
Figure 2: Defining continuity using the notion of open neighborhoods.
Definition 1.6: Continuity at a point (Euclidean space)
A function \(f: \mathbb{R}^n \to \mathbb{R}^m\) is said to be continuous at \(x_0 \in \mathbb{R}^n\) if for any open neighborhood \(V\) of \(f(x_0)\), \(\exists\) an open neighborhood \(U\) of \(x_0\), such that, \(f(U) \subset V\).
Or equivalently,
Definition 1.7: Continuity at a point (Euclidean space)
A function \(f: \mathbb{R}^n \to \mathbb{R}^m\) is said to be continuous at \(x_0 \in R^n\) if for any open neighborhood \(V\) of \(f(x_0)\), \(f^{-1}(V)\) is an open neighborhood of \(x_0\), where the inverse(or pre-image) of \(f\) is defined as \( f^{-1}(V) = \{x \in X \mid f(x) \in V\}\) for \(V \in R^m\).
This definition does not use distances but the concept of open neighborhoods. Therefore, one could generalize this definition directly to topological spaces. However, continuity at a point is not a very useful concept for topological spaces. Hence, we present the topological definition of a continuous function. In the Euclidean (or a metric) space, a continuous function is defined as a function which is continuous at all points of its domain.
Definition 1.8: Continuous function (Topological space)
A function \(f:X \to Y\) between two topological spaces \(X, Y\)
is said to be continuous if for every open set
\(V \in Y\), \(f^{-1}(V)\) is open in \(X\).
In the next article of the series we shall continue our discussion towards understanding topological manifolds, differentiable manifolds and Riemannian manifolds.
References
[1] John M. Lee. Introduction to Topological Manifolds. Graduate Texts in Mathematics. Springer, 2011.[2] P.A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008.
