A theory which satisfies all the Eilenberg-Steenrod axioms with the possible exception of the long exact sequence of a pair axiom, as well as a certain additional continuity condition.
Aleksandrov-Čech Cohomology
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References
Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, p. 68, 1988.Referenced on Wolfram|Alpha
Aleksandrov-Čech CohomologyCite this as:
Weisstein, Eric W. "Aleksandrov-Čech Cohomology." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Aleksandrov-CechCohomology.html