Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be the real numbers. Is the following topology stronger than the cofinite topology? And is $X$ then compact?

$T = \{ U \subset\mathbb{R}\:|\:0\notin U\text{ or }\mathbb{R}\setminus U\text{ finite}\} $

share|cite|improve this question
What is $X$? How does finiteness of $X\setminus \mathbb R$ affect $U$? What is a compact topology? It is homework? – Ilya Jan 11 '13 at 14:28
And does "stronger" mean having more or less open sets? – Chris Eagle Jan 11 '13 at 14:31
More open sets.. Because now you have finite open sets right? Every set without $0$ belongs to the topology (along with all cofinite sets). – omar Jan 11 '13 at 14:34
@Chris: Where have you seen it used to mean fewer open sets? – Brian M. Scott Jan 11 '13 at 14:36
@ChrisEagle: I like this "especially analysts"! I always knew one shall be careful with these people – Ilya Jan 11 '13 at 14:42
up vote 7 down vote accepted

You can write this as the union of the cofinite topology and the union of all sets not containing $0$, so it is stronger. It is also compact. Every open covering must contain an element containing $0$ and this element will have a finite complement. Supply the details yourself.

share|cite|improve this answer
So the proof that the above given space X is compact would be: Let $\mathcal{U}$ be an opencover of $X$, then there exists a $U_0 \in \mathcal{U}$ which contains $0$ such that $X\setminus{U_0}$ is finite. Then, for every $x \in X\setminus{U_0}$ there is a $U_x \in \mathcal{U}$ containing x. The union $U_0 \cup \{U_x : x \in X\setminus{U_0} \}$ is a finite open cover of X. So X is compact. – omar Jan 11 '13 at 18:02
@omar Exactly.${}$ – Michael Greinecker Jan 11 '13 at 18:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.