Consider with the topology generated by , which is a T0 space. The derived set of is , which is open.
Originally Posted by austinmohr
Ack - a typo. The set is neither open nor closed in this topology. The counterexample still works, however.
If this is Lynn Steen’s Left Order Topology, then isn’t it true that is a limit point of
If so, then is in the derived set of
If that is so, what is its complement?
If this is Lynn Steen’s Left Order Topology, then isn’t it true that is a limit point of
If so, then is in the derived set of
If that is so, what is its complement?
We may be using slightly different definitions. I use "derived set" to mean the set of all accumulation points of a set . Hence, if is singleton, it cannot be contained in its own derived set by definition.
We may be using slightly different definitions. I use "derived set" to mean the set of all accumulation points of a set . Hence, if is singleton, it cannot be contained in its own derived set by definition.
You are correct. I was not thinking of a singleton set.
In fact, I don't know what I had in mind.
Maybe I really don't know.
Last edited by Plato; November 24th 2009 at 03:25 AM.