Coarser connected topologies and non-normality points

We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is: Question 0.0.1 . When does a space have a coarser connected topology with a nice topological property?;We will discuss some results when the property is Hausdorff and prove that if X is a non-compact metric space that has weight at least c, then it has a coarser connected metrizable topology.;The second topic is concerned with the following question: Question 0.0.2. When is a point y ∈ betaX X a non-normality point of betaX X?;We will discuss the question in the case that X is a discrete space and then when X is a metric space without isolated points. We show that under certain set-theoretic conditions, if X is a locally compact metric space without isolated points then every y ∈ betaX X a non-normality point of betaX X.