Separation properties and n-point topological extensions

A topological extension of a topological space (X, j) is a topological space (X*,j*) containing (X,j) as a dense subspace. Two topological extensions (X*,j*), and (X*₁,j*₁) of (X,j) are said to be equivalent if there is a homeomorphism h:(X*,j*) +(X*₁,j*₁) such that hlX = id_x.