Compactness of A in R2 with Standard Topology: Tychonoff's Theorem Applied

[jangoc44]

1. The question is to show whether A is compact in R2 with the standard topology. A = [0,1]x{0} U {1/n, n\in Z+} x [0,1]

3. If I group the [0,1] together, I get [0,1] x {0,1/n, n \in Z+ }, and [0,1] is compact in R because of Heine Borel and {0}U{1/n} is compact since you can show that every cover has a finite subcover. Now, if you take the product of two compact sets will they still be compact? Tychonoff's theorem says product of compact spaces is compact, but I'm not too sure if it applies here.

Thanks in advance.

 

[Martin Rattigan]

A = [0,1]x{0} U {1/n, n\in Z+} x [0,1]

Where is the principal operation in the expression for A. If A\subset \mathbb{R}_2 it can't have two product operations, so I'd go for the union as it stands. But this doesn't look too likely, and doesn't seem to tie up with what you say in 3., so I suspect there may be a typo.

 

[jangoc44]

Sorry I meant ([0,1]x{0}) U ({1/n}x[0,1]).

 

[Martin Rattigan]

OK - it was your reference to {0}U{1/n} that was throwing me a bit. Look at the points on {0}x[0,1] and go direct from the definition. (That assumes you define compact as closed and bounded, otherwise you will no doubt have proved these properties in your course from whatever definition you use.)

 

[Martin Rattigan]

I'm off to bed now so should you have any problems hopefully someone else will pick up the thread.