The lexicographic ordered products and the usual Tychonoff products

The usual Tychonoff product space of arbitrary many compact (ω-bounded) spaces is well-known to be also compact (ω-bounded). In this paper, we compare the lexicographic ordered topologies on some products of ordinals with the Tychonoff product topologies. We see:• xicographic ordered space ω 1 ω is ω-bounded. xicographic ordered space ω 1 ω + 1 is not ω-bounded. nd β are ordinals with β < α , then the lexicographic ordered space [ 0 , β ] ω is a subspace of the lexicographic ordered space α ω , thus the lexicographic ordered space 2 ω is a subspace of the lexicographic ordered space 3 ω . xicographic ordered space 2 ω + 1 is not a subspace of the lexicographic ordered space 3 ω + 1 . l n < ω with 2 ⩽ n , the lexicographic ordered space n ω is homeomorphic to the Cantor set. xicographic ordered space 2 ω + 1 is not metrizable. xicographic ordered spaces 2 ω + 1 and 3 ω + 1 are not homeomorphic. xicographic ordered topology on ω × 2 ω coincides with its usual Tychonoff product topology. xicographic ordered topology on ω ω is strictly weaker than its usual Tychonoff product topology. xicographic ordered topology on ω × ω × ω 1 is strictly weaker than its usual Tychonoff product topology. xicographic ordered topology on ω × 2 × 3 × 4 × ( ω 1 + 1 ) coincides with its usual Tychonoff product topology.