# Joyal's CatLab Introduction to topology

## Introduction to topology

### History

Topology is a ‘subfield’ of mathematics which attempts to capture and study the minimal structure of space. General topology seeks to do this through axiomatizing the idea of neighborhoods of a point. The first definition of a topological space seems to be due to Hausdorff1 in 1914. We will take a short detour and first introduce metric spaces and use it to explain the intuition behind the general axioms for a topological space.

### A detour: introduction to metric spaces

The concept of a metric space is actually a bit older than a general topological space, albeit only by approximately 8 years. A metric space is the following: $(X,d)$, where $X$ is the set of points of our space, and $d: X \times X \rightarrow \mathbb{R}$ is a metric. A metric intuitively ‘measures’ the distance between two points, and must satisfy the following axioms:

1. Non-negativity: $d(x,y) \geq 0$, so that the ‘distance’ is never negative.

2. Coincidence axiom: $d(x,y) = 0$ if and only if $x=y$, so that(with 1.) different points always have a positive distance between them.

3. The triangle inequality: If $x,y$ and $z$ are three points of $X$(they may be the same points), we should have that $d(x,y) + d(y,z) \geq d(x,z)$

4. Symmetry: $d(x,y) = d(y,x)$ for every point $x$ and $y$ of $X$.

So what do we want a neighborhood $U$ of a point $x$ to be? Well, intuitively it should contain the ‘neighbors’ of $x$. That is there should be a positive real number $r$ where for any $y \in X$, the condition $d(x,y) \lt r$ implies that $y \in U$. Let’s introduce a piece of notation to facilitate this. Let $B(x,r)$ be the ‘open ball’ of radius $r$ based at $x$, that is $B(x,r) = \{ y \in X : d(x,y) \lt r \}$. Then our intuition says we should define a ‘neighborhood’ of $x$ to be a set $U$ containing $x$, where there is a positive $r$ with $B(x,r) \subset U$.

Now we define an open subset of $X$ to be a set $U$ where $U$ is a neighborhood of every point in $U$, that is if $u$ is a point of $X$ with $u \in U$, then we can find a $r_u \gt 0$ with $B(u,r_u) \subseteq U$. Notice that the family of open subsets of $X$ has a few nice properties:

1. $X$ is an open subset of $X$, by taking any positive $r_x$ for each $x$ we see $X$ is a neighborhood of every point in $X$.

2. $\emptyset$ is an open subset of $X$, by the triviality that it has no points, so it is a neighborhood of every point inside of it.

3. If $U$ and $V$ are two open subsets of $X$, then $U \cap V$ is also an open subset of $X$. Let $w \in U \cap V$, so that $w \in U$ and $w \in V$. By the former, we get a $r_1 \gt 0$ with $B(w,r_1) \subseteq U$ and the latter gives $r_2 \gt 0$ with $B(w,r_2) \subseteq V$. We see that $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_1)$ and $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_2)$, so that $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq U$ and $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq V$, but then $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq U \cap V$. So $U \cap V$ is a neighborhood of every point inside of it, thus an open subset of $X$.

4. If $\{ U_i\}_{i \in I}$ is an arbitrary collection of open sets, then the union of them $\cup U_i$ is also open. The proof is left as an exercise to the reader.

It is these 4 properties which we take as the basic properties of open sets.

### Definition of a topological space

A topological space is a pair $(X, \tau)$2, where $\tau$ is a collection of subsets of $X$ satisfying the following properties:

1. $X \in \tau$ and $\emptyset \in \tau$

2. If $\{ U_i\}_{i \in I} \subset \tau$, then $\cup U_i \in \tau$

3. If $U,V \in \tau$ then $U \cap V \in \tau$

The elements of $\tau$ are called open subsets of $X$, or if the context is clear they are often merely called open sets. A set is called closed if it’s complement is open.

An non-empty set $S$ with two or more elements always has at least two topologies: the trivial topology $(S,\tau)$ and the discrete topology $(S,\tau')$. The former is $\tau = \{ \emptyset, S\}$ and the latter has $\tau' = \mathcal{P}(S)$ the power set of $S$. If $S$ has one element or no elements, these are the same.

There is almost nothing true one may say about a topological space without requiring additional properties!

1. Hausdorff’s axiomatization is not the modern one, it seems to include the Hausdorff condition as an axiom.

2. You will probably never see the notation $(X,\tau)$ for a topological space again while reading and working, but a bit of completeness at this point is nice and can clarify certain discussions(like comparing different topologies on the same space).

Created on September 1, 2012 at 23:19:52 by jstalfos