A new property called P β -connectedness is introduced which is stronger than connectedness and equivalent to pre-connectedness. The properties of this notion are explored and its relationship with other forms of connectedness, for example hyperconnectedness etc. are discussed. Locally pre-indiscrete spaces are defined as the spaces in which pre-open sets are closed. In such spaces connectedness becomes equivalent to pre-connectedness and hence to P β -connectedness, and semi-connectedness becomes equivalent to P β - connectedness. The notion of locally P β -connected space is introduced. The behavior of P β -connectedness under several types of mappings is investigated. An intermediate value theorem is obtained.
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