Some Cardinal and Geometric Properties of the Space of Permutation Degree

Abstract

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

Some network-type properties of the space of G-permutation degree

“…The study of the influence of normal, weakly normal and seminormal functors to topological and geometric properties of topological spaces, in particular to the cardinal properties (density, weak density, local density, tightness, set tightness, T -tightness, functional tightness, mini-tightness), has been developed in recent investigations (see [1,2,3,4,6,7,9,10,11,12,14,15,16,17]). For example, in [2] it was proved that the exponential functor of finite degree preserves the functional tightness and minimal tightness of compact sets, while in [11] a similar investigation (related to the T −tightness, set tightness, functional tightness, mini-tightness) was done for the functor SP n G of G-permutation degree.…”

Some Tightness-Type Properties of the Space of Permutation Degree

“…In recent researches a number of authors was interested in the behaviour of certain cardinal invariants under the influence of various covariant functors. For example, in [5,11,12,14,19,20] the authors investigated several cardinal invariants under the influence of some weakly normal, seminormal and normal functors.…”

“…In [11,12] some cardinal and geometric properties of the space of permutation degree SP n X have been discussed. It is proved that if the product X n has some Lindelǒf-type properties, then the space SP n X also has these properties.…”

Tightness-Type Properties of the Space of Permutation Degree

“…In recent research, a number of authors have investigated the behaviour of certain cardinal invariants under the influence of various covariant functors. For example, in [3][4][5][6][7][8] the authors investigated several cardinal invariants under the influence of weakly normal, seminormal, and normal functors.…”

“…In [4,5], the authors discussed certain cardinal and geometric properties of the space of the permutation degree SP n X. They proved that if the product X n has some Lindel ǒf-type properties, then the space SP n X has these properties as well.…”