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This article deals with a continuous closed 1-form defined on a CW-complex.In particular, we show Lusternik-Schnirelmann type theory on continuous closed1-forms which is related to gradient-like flows. M.Farber defined a continuousclosed 1-form and a category with a respect to a cohomology class andconstructed a Lusternik-Schnirelman Theory for smooth closed 1-forms. We denoteabout a definition of a zero-point of continuous closed 1-form and generalizethis theory to continuous closed 1-forms. It shows a relation between thenumber of a zero-point of continous closed 1-form and the category with arespect to a cohomology class, through a dynamical system on a original space,homoclinic cycle.

Closed 1-form on topological spaces, and Lusternik-Schnirelman type Category