- #1
kathrynag
- 598
- 0
Homework Statement
If [tex]E_{1}[/tex],...[tex]E_{n}[/tex] are compact, prove that E=[tex]\cup^{n}_{i=1}[/tex][tex]E_{i}[/tex] is compact.
Homework Equations
The Attempt at a Solution
A set E is compact iff for every family {[tex]G_{\alpha}[/tex]}[tex]_{\alpha\in}A[/tex] of open sets such that E[tex]\subsetU_{\alpha\in}A G_{\alpha}[/tex]
Let [tex]G_{\alpha}[/tex]=[tex]E_{n}[/tex].
Let E=(i,n)
If i<x<n, there is a positive integer n such that [tex]E_{n}[/tex]<x, hence x[tex]\in[/tex][tex]G_{n}[/tex] and E[tex]\subset[/tex][tex]G_{n}[/tex].
Not quite sure about this one.