We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is: 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 the cardinality of the continuum, then it has a coarser connected metrizable topology. The second topic is concerned with the following question: When is a point of the Stone-Cech remainder of a space a non-normality point of the remainder? 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 point in the Stone-Cech remainder is a non-normality point of the remainder.