SlipEternal
Junior Member
- Joined
- Jan 4, 2012
- Messages
- 114
Is it possible for a space to be compact, Hausdorff, totally disconnected, first countable, and nowhere locally metrizable?
This problem came up as I was trying to understand the topological characterization of the Cantor set, and since it is not part of any topology material, my professor suggested that I look into it on my own. I almost had myself convinced that if you had a space that was Compact, Perfect, Totally disconnected, T0, and first countable, then there must be some nonempty open set that is homeomorphic to the Cantor set. Unfortunately, I found a flaw in my proof.
So, I am wondering if anyone knows if there is an example of such a space that is compact, hausdorff, totally disconnected, first countable, and nowhere locally metrizable.
This problem came up as I was trying to understand the topological characterization of the Cantor set, and since it is not part of any topology material, my professor suggested that I look into it on my own. I almost had myself convinced that if you had a space that was Compact, Perfect, Totally disconnected, T0, and first countable, then there must be some nonempty open set that is homeomorphic to the Cantor set. Unfortunately, I found a flaw in my proof.
So, I am wondering if anyone knows if there is an example of such a space that is compact, hausdorff, totally disconnected, first countable, and nowhere locally metrizable.