Register to reply

R with the cocountable topology is not first countable

by mrbohn1
Tags: cocountable, countable, topology
Share this thread:
Oct20-08, 05:21 PM
P: 97
1. The problem statement, all variables and given/known data

(a) Prove that R, with the cocountable topology, is not first countable.

(b) Find a subset A of R (with the cocountable topology), and a point z in the closure of A such that no sequence in A converges to z.

2. Relevant equations

(The cocountable topology on R has as its closed sets all the finite and countable subsets of R).

3. The attempt at a solution

I'm not really sure how to prove this. I know that if I could find a set and point meeting the conditions in part (b), then that would prove part (a), as in a first countable space X a point x belongs to the closure of a subset A of that space if and only if there is a sequence of points of A converging to x. However, I assume that I am expected to prove that R is not first countable with this topology some other way for part (a).

Either way, I'm stuck!
Phys.Org News Partner Science news on
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Oct20-08, 06:30 PM
P: 336
When you say *first* countable do you mean isomorphic to Z? Maybe it would be easier to show that the cocountable topology on Z isn't countable.
Oct21-08, 01:26 PM
P: 97
(from wikpedia): "a space, X, is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point, x, in space X there exists a sequence, U1, U2, of open neighborhoods of x such that for any open neighborhood, V, of x, there exists an integer, i, with Ui contained in V."

Oct21-08, 03:48 PM
Sci Advisor
HW Helper
P: 2,020
R with the cocountable topology is not first countable

Suppose there's a countable basis at 0, say {U_1, U_2, ...}. What is the intersection of all the U_i?

Register to reply

Related Discussions
K topology strictly finer than standard topology Calculus & Beyond Homework 5
I need help proving that the cross product of 2 countable sets is countable. Calculus & Beyond Homework 6
A metric space having a countable dense subset has a countable base. Calculus & Beyond Homework 2
Countable But Not Second Countable Topological Space General Math 3
Geometric Topology Vs. Algebraic Topology. General Math 1