(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if {K_n} is a decreasing family of compact connected sets in a metric space, then their intersection is connected as well. Illustrate with an example why 'compact' is necessary instead of just 'closed'.

3. The attempt at a solution

Well, I have a example for the second part of the question. Consider F_n = R²\{(x,y): -n<y<n, -1<x<1}. Then each F_n is closed and (path-)connected, but their intersection is the plane separated in half along the y axis by this open band of width 2, which is not connected.

For the first part though, I can visualize why it's true for simple examples, but I don't know how to approach a general proof.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Metric space topology problem

**Physics Forums | Science Articles, Homework Help, Discussion**