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Introduction to topology

Introduction to topology


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)(X,d), where XX is the set of points of our space, and d:X×Xd: 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)0d(x,y) \geq 0, so that the ‘distance’ is never negative.

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

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

  4. Symmetry: d(x,y)=d(y,x)d(x,y) = d(y,x) for every point xx and yy of XX.

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

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

  1. XX is an open subset of XX, by taking any positive r xr_x for each xx we see XX is a neighborhood of every point in XX.

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

  3. If UU and VV are two open subsets of XX, then UVU \cap V is also an open subset of XX. Let wUVw \in U \cap V, so that wUw \in U and wVw \in V. By the former, we get a r 1>0r_1 \gt 0 with B(w,r 1)UB(w,r_1) \subseteq U and the latter gives r 2>0r_2 \gt 0 with B(w,r 2)VB(w,r_2) \subseteq V. We see that B(w,textrmmin{r 1,r 2})B(w,r 1)B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_1) and B(w,textrmmin{r 1,r 2})B(w,r 2)B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_2), so that B(w,textrmmin{r 1,r 2})UB(w, \textrm{min}\{ r_1,r_2\}) \subseteq U and B(w,textrmmin{r 1,r 2})VB(w, \textrm{min}\{ r_1,r_2\}) \subseteq V, but then B(w,textrmmin{r 1,r 2})UVB(w, \textrm{min}\{ r_1,r_2\}) \subseteq U \cap V. So UVU \cap V is a neighborhood of every point inside of it, thus an open subset of XX.

  4. If {U i} iI\{ U_i\}_{i \in I} is an arbitrary collection of open sets, then the union of them U i\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,τ)(X, \tau)2, where τ\tau is a collection of subsets of XX satisfying the following properties:

  1. XτX \in \tau and τ\emptyset \in \tau

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

  3. If U,VτU,V \in \tau then UVτU \cap V \in \tau

The elements of τ\tau are called open subsets of XX, 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 SS with two or more elements always has at least two topologies: the trivial topology (S,τ)(S,\tau) and the discrete topology (S,τ)(S,\tau'). The former is τ={,S}\tau = \{ \emptyset, S\} and the latter has τ=𝒫(S)\tau' = \mathcal{P}(S) the power set of SS. If SS 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,τ)(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