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A subset N of a topological space is defined to be a θ-neighborhood of x if there exists an open set U such that x∈U⫅C1   U⫅N. This concept is used to characterize the following types of functions: weakly continuous, θ-continuous, strongly θ-continuous, almost strongly θ-continuous, weakly δ-continuous, weakly open and almost open functions. Additional characterizations are given for weakly δ-continuous functions. The concept of θ-neighborhood is also used to define the following types of open maps: θ-open, strongly θ-open, almost strongly θ-open, and weakly δ-open functions.