The invariance of domain theorem for the finite product of connected linearly ordered topological spaces

We prove that for every positive integer n , every continuous injective function from the product of n -many connected linearly ordered topological spaces with at least two points and no endpoints into the finite product of n -many connected linearly ordered topological spaces is an open map. By applying this theorem, we also prove that every connected cancellative topological semigroup on the finite product of connected linearly ordered topological spaces is locally Euclidean. • We prove the invariant of domain theorem for the finite product of connected linearly ordered topological spaces. • We prove that every cancellative topological semigroup on the finite product of connected linearly ordered topological spaces without endpoints is locally Euclidean. • We prove that every cancellative topological monoid on the finite product of connected linearly ordered topological spaces without endpoints is a Lie group.