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Metric and topological spaces - part 1

Last summer our (now) Academic Secretary Stijn Hanson organised a small study group on Topology, and as part of this I gave a few talks. I thought I would share more information on this area as it is very insightful in characterizing many of the most important aspects of the general notion of space, and continuity, and highly recommended to those with an interest in Algebra or Analysis. So, starting with this introduction, I will be writing a series of articles on two of my favourite topics, Metric spaces and Topology.

Part 1: Meet the spaces

Firstly, for the uninitiated, I will briefly introduce Metric and Topological Spaces. Both are both abstractions of our common notions of space (a.k.a. n-dimensional Euclidean space, \mathbb{R}^n), with Topological Spaces capturing the notion of continuity, and closed and open sets, whereas Metric Spaces center around the notion of distance.

Metric Spaces

A Metric Space is a non-empty set X assigned a map \rho : X \times X \rightarrow \mathbb R, called a Metric on X, which denotes the distance between two points. We will write this as (X, \rho). The map \rho must satisfy the following 3 properties of distance:

  • \forall x, y \in X, \rho (x, y) = 0 \Leftrightarrow x = y  (faithfulness)
  • \forall x, y \in X, \rho (x, y) = \rho (y, x)  (symmetry)
  • \forall x, y, z \in X, \rho (x, z) \leq \rho (x, y) + \rho (y, z)  (triangle inequality)

We can see the intuition for these by looking at the example of 2 dimensional Euclidean space, that is,  \mathbb{R}^2 under the metric,

 \rho_2 ((x_0, y_0), (x_1, y_1)) = \lVert (x_0, y_0) - (x_1, y_1) \rVert = \sqrt{(x_0 - x_1)^2 + (y_0 - y_1)^2}

Faithfulness simply says that that the distance from a point to itself, and if the distance between two points is equal, then they are infact the same point. Symmetry says that when we talk about the distance between two points, it does not matter in which order we specify them. The triangle inequality is a bit more interesting, and is based on the fact from elementary geometry that the length of the hypotenuse of any triangle is less than or equal to the combined length of the other two sides. It also generalise the ubiquitous triangle inequality of Real Analysis: \forall x, y \in \mathbb{R}, \lvert x - y \rvert \leq \lvert x \rvert + \lvert y \rvert .

Topological Spaces

Topological spaces arise from the realization that many useful properties of space depend not on distance but a notion of open and closed sets, and are defined as follows.

A Topological Space is a non-empty set X along with a system \tau \subseteq \mathcal{P}(X) of subsets of X, such that

  • X, \emptyset \in \tau
  • \tau is closed under union: 

    \left( \forall \lambda \in \Lambda, U_{\lambda} \in \tau \right) \Rightarrow \bigcup \limits_{\lambda \in \Lambda} U_{\lambda} \in \tau

  • \tau is closed under finite intersection: 

    \left(U_1, \ldots, U_n \in \tau\right) \Rightarrow \bigcap \limits_{k=1}^n U_n \in \tau

We write this as (X, \tau). We call every set U \in \tau open, and X \setminus U closed.

We can see how this definition arises by looking at open and closed sets in (\mathbb{R}^2, \rho_2) and other metric spaces. Where \epsilon \in \mathbb{R}^+ we define the open ball of radius \epsilon  around x \in X  as

B_{\epsilon}(x) = \left\{ y \in X : \rho(x, y) \lt \epsilon \right\}

and the closed ball of radius  \epsilon around  x \in X as

B_{\epsilon}[x] = \left\{ y \in X : \rho(x, y) \leq \epsilon \right\}

In \mathbb R the open ball B_{\epsilon}(x) is simply the open interval (x-\epsilon, x+\epsilon), whereas the closed ball B_{\epsilon}[x] is simply the closed interval [x-\epsilon, x+\epsilon], and in general the open and closed balls play much the same role as these two types of intervals do in Real Analysis.

open and closed

It can also be seen that in \mathbb{R}^2 that the open ball is just a circle of radius \epsilon centred at x excluding its boundary, whereas the closed ball is just a circle of radius \epsilon centred at x including its boundary.We can extend this to a general definition of an open set in a metric space by saying that all open balls are open, and that any union of open sets is open. In general, a set is closed in a metric space if its complement is closed.But then the open sets of a metric space X form a topology, \tau_X = \left\{ U \subseteq X : U \: \text{ is open in } \: X \right\}

Notes and Sources

If you are interested in looking further into the topic, the notes from the original study group are below:

Creative Commons Licence This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. These notes and the article were based upon:

If you like these topics then you should look at the following York courses:

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