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Topologies on R of All Possible Cardinalities

13th February 2009, 10:29 am

In this guest column, David Thompson (Williams ’11) observes that there are topologies on R of all cardinalities from 2 to 2^{|R|}, assuming the Continuum Hypothesis, a result from the first topology tutorial session with his tutorial partner Andrew Lee and me. In my topology tutorial, the 12 students meet weekly with me in groups of two or three to present and discuss the material.

P.S. In the subsequent comment, fellow student Rob Silverstein generalizes part of the result to any space X.

On the Cardinalities of Topologies of R by David Thompson

In discussing the definition of a topology on a general set, a natural question arises regarding the sizes of allowable topologies.  A topology is, after all, a collection of subsets obeying certain properties, and the size of this collection can vary drastically. First recall the definition of a topology:

A topology T on a set X is a collection of open sets U such that:

1.) X, Ø \in T,

2.) T is closed under arbitrary unions and finite intersections.

Turning our attention to the real line, R, we see that the smallest possible topology is one containing the two members mandated by property 1; namely, T = {R, Ø}.  We now claim that there is a topology on R containing three members.  Let T = {R, Ø, {0}}.  Looking at all possible unions and intersections of members of T, we see that T is indeed a topology.  Similarly, we can generate a topology of any finite or countable cardinality by taking:

T = {R, Ø, {0}, (-1,1), (-2,2),  …(-n,n)} for some n\inN (to achieve the same cardinality as N simply let n range through all of N).

In fact, this method even provides us with a topology with the same cardinality as the continuum—merely allow n to range through the positive reals instead of the positive integers.

Finally, we introduce the notion of the discrete topology on a set X.  The discrete topology is the collection of all subsets of X.  In the case of the real line, we have the collection of all subsets of R, which has cardinality 2^{|R|}.  Since the discrete topology is the largest topology we can have on R, it follows there is no topology with cardinality greater than 2^{|R|}.  What does this imply about the allowable sizes of topologies on R?  Assuming the Continuum Hypothesis, which states that there is no set with cardinality strictly between that of N and R, or R and 2^{|R|}, we see that we can designate a topology on R of every size from 2 to 2^{|R|}.  

 

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2 Comments

  1. Rob Silversmith:

    Can we generalize to arbitrary sets?

    Let X be a set with cardinality K. Then given k≤K, there exists some subset A of X with cardinality k. Well order A. The topology T_A, consisting of all sets {x<a} for a in A, together with X, has cardinality k+1.

    13 February 2009, 1:12 pm

  2. Jake Levinson:

    I don’t know if you even need to use well-ordering. Any descending chain of nested subsets of a set X (assuming it includes X itself and the empty set) will be a topology on X, so you can just use any old sequence of nested subsets, of whatever length you want, finite or infinite, countable or uncountable (as long as sufficiently many such subsets exist in X.)

    Thanks Jake. Actually I edited Rob’s comment to a “well” ordering mainly for the trivial reason of making sure that the empty set was included and because the easiest proof I know that every set can be ordered is the proof that every set can be well-ordered.—FM

    14 February 2009, 2:28 am

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