On hereditary properties of extremally disconnected frames and normal frames

Perfect locales and localic real functions

“…The results above have interesting counterparts in the dual case of extremally disconnected locales (in the vein of the approach in [10] and [8]).…”

Certain Strong and Weak Types of Paracompact Maps

“…Then, 𝕂 is a paracompact set in 𝕄 by Theorem 2.3 part (10). This implies ℒ −1 (𝕂) is paracompact in 𝕎 owing to ℒ is a paracompact map which follows ℒ −1 (𝕂) is fully normal because of Theorem 2.3 part (11). Hence, is a fully normal map.…”

“…Proof: Let ℒ: 𝕎→𝕄 be a fully normal map such that 𝕎 is a T 1 -space and 𝕄 is a Hausdorff space. Assume that 𝕂 is a paracompact set in 𝕄. Thus,𝕂 is a Hausdorff subspace of 𝕄 because 𝕄 is a Hausdorff space and so, 𝕂 is a fully normal subset of 𝕄 by Theorem 2.3 part (11). Now, ℒ −1 (𝕂) is fully normal in 𝕎 due to ℒbeing fully normal.…”

“…Assume that 𝕂 is a fully normal set in 𝕄. So, 𝕂 is a fully T 4 set in 𝕄 by Theorem 2.2 part (11). thus, ℒ −1 (𝕂) is fully T 4 in 𝕎 due to ℒ being a fully T 4 map.…”