# topology- compactness, interior=empty

• May 4th 2011, 01:42 PM
tom007
topology- compactness, interior=empty
I need help with this exercise..(Thinking)

Let $\{ X_\alpha\}_{\alpha \in I}$ be a collection of topological spaces. $X=\prod_{\alpha \in I}X_\alpha$
1)prove: If an infinite number of $X_\alpha$ are non-compact, then any compact subset in $\prod_{\alpha \in I}X_\alpha$ has empty interior. 2)is there a need for every compact set $K\subseteq X$to be dense nowhere in $X$( $Int \overset{-}{K}=\emptyset$)?

..I look at the product $\prod_{n \in \mathbb N}X_n$ where all $X_n$ are equal to set $\mathbb R \cup \{c\}$ $(c \notin \mathbb R$) with a topology whose basis consists of sets in form of $\langle a,b \rangle \cup \{c\}, a,b\in\mathbb R$

Thank you!
• May 4th 2011, 02:39 PM
Tinyboss
Basis elements in the product topology are the entire space in all but finitely many factors. So if infinitely many factor spaces are non-compact, then no basis element is contained in a compact set.
• May 5th 2011, 12:48 AM
tom007
mm..cool :)
and that means that it has empty interior..tnx