In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.

The concept of soft ideal was first introduced by Kandil et al. [13]. In 1999, Molodtsov [22] introduced the concept of soft sets as a general mathematical tool for dealing with uncertain objects. The concept of generalized closed soft sets in soft topological spaces was introduced by Kannan [15] in 2012. The notions of supra soft topological space were first introduced by Kandil et al. [14]. I...

A topological space is called paracompact (see [2 J) if (i) it is a Hausdorff space (satisfying the T2 axiom of [l]), and (ii) every open covering of it can be refined by one which is "locally finite" ( = neighbourhood-finite; that is, every point of the space has a neighbourhood meeting only a finite number of sets of the refining covering). J. Dieudonné has proved [2, Theorem 4] that every se...

We study weakly continuously Urysohn spaces, which were introduced in [Z]. We show that every weakly continuously Urysohn w∆-space has a base of countable order, that separable weakly continuously Urysohn spaces are submetrizable, hence continuously Urysohn, that monontonically normal weakly continuously Urysohn spaces are hereditarily paracompact, and that no linear extension of any uncountabl...

This result is also due independently to G. Hector [3], who has shown how useful it can be in understanding the geometry of certain foliated manifolds. In such applications one sometimes needs a form of this theorem which applies to foliated subspaces, for example a minimal subset of a foliation. In fact our proof goes through unaltered in the situation where M is a locally compact, paracompact...

In [6], Yoshikazu Yasui formulates some results on relative countable paracompactness and poses some questions. Like it is the case with many other topological properties [1], countable paracompactness has several possible relativizations. Thus a subspace Y ⊂ X is called countably 1-paracompact in X provided for every countable open cover U of X there is an open cover V of X which refines U and...

Let f : X → Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space. We consider the following property of continuous maps g : X → Ik = [0, 1], where 1 ≤ k ≤ ω: each g(f(y)), y ∈ Y , is at most n-dimensional. It is shown that all maps g ∈ C(X, In+1) with the above property form a dense Gδ-set in the function space C(X, I n+1 ) equipped with the source limitation topolo...