Abstract
Let M be a non-empty proper subset of a space X. The discrete extension space X M is de fined by declaring that any element of X\M is isolated and any element of M has the same open neighborhood as in X. We study the relationship between a space X and a discrete extension X-M of it according to the properties C-normality, L-normality, CC-normality, S-normality, countable normality, and others. We use a discrete extension space to solve an open problem regarding C-normality.