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The mathematical structure of cyberworlds is clarified based on the duality of homology lifting property and homotopy extension property. The duality gives bottom-up and top-down methods to model, design and analyze the structure of cyberworlds. The set of homepages representing a cyberworld is transformed into a state finite machine. In development of the cyberworld, a sequence of finite state machines is obtained. This sequence has homotopic property. This property is clarified to map a finite state machine to a simplicial complex. Wikipedia, bottom-up network construction and top-down network analysis are described as examples.