A topological space is locally compact then there is an open base at each point has all of its set with compact closure, how can i prove it?
It seems very meaningful when the topological space is metrizable and the topology is the topology induced by the metric, but i could not prove it for general topological spaces.
I am not familiarized with Hausdorff spaces.
What i tried for now is starting from the definition of locally compact as the topological space whose ever point has a neighbourhood with compact closure. I picked (for each point) this neighbourhood, and created a class with all open sets contained in that neighbourhood, then i thought that this class may be the open base at the analized point whose all sets have compact closure, but i could not prove that this class is an open base at point, have someone an idea? Or other path to prove it, or yet is this claiming false?