A Study On Rectifiable Spaces And Their Remainders In Compactifications And Related Conclusions

Recall that a topological group G is a group with a topology such that the product map ofG×G into G is jointly continuous and the inverse map onto itself associating x1with x∈Gis continuous. A topological space G is said to be a rectifiable space provided that there are asurjective homeomorphism ψ: G×G→G×G and an element e∈G such that π1ψ=π1and ψ(x, x)=(x, e) for each x∈G, where π1: G×G→G is the projection to the firstcoordinate. It is well known that every topological group is a rectifiable space.In1996, A.S. Gul’ko proved that a topological space G is rectifiable if and only if thereare two continuous mappings p: G2→G, q: G2→G such that for any x∈G, y∈G andsome e∈G the next identites hold: p(x, q(x, y))=q(x, p(x, y))=y, q(x, x)=e. If G is arectifiable space and a map p: G2→G satisfies the conditons of the theorem, then we dnotep(x, y) by x· y and p(A, B) by A· B, where A G, B G.In1958, M. Henriksen and J.R. Isbell showed that a space X is of countable type if andonly if the remainder in any compactification of X is Lindelo¨f. In recent years, many results ontopological groups and their reminders proved by A.V. Arhangelskii, Chuan Liu and Shou Lin,Liangxue Peng based on the above result.In2010, A.V. Arhangel’skii and M.M. Choban posed the following question: Is everyrectifiable p-space with a countable Souslin number Lindelo¨f? What if we assume the space tobe separable? Separable and locally compact? By the studying properties of rectifiable spaces,we give a positive answer to the above question of the case of rectifable p-space with a countableSouslin number. Thus every rectifiable p-space with a countable Souslin number is Lindelo¨f.In2011, Fucai Lin and Rongxin Shen posed the following question: Let G be a rectifiablespace. If C, F are compact and closed subsets of G respectively, is C· F or F· C closed in G?By the studying properties of rectifiable spaces, we show that C· F is closed in G if C, F arecompact and closed subsets of a rectifiable space G, respectively.In2010, Liangxue Peng and Yufeng He gave a summary on properties of a remainder ofa non-locally compact topological group G in a compactification bG makes the remainder andthe topological group G are all separable and metrizable. In this paper, we point out that arectifiable paracompact space has a similar property. Thus if a non-locally compact rectifiableparacompact space G has a comopactification bG such that the remainder bG\G∈P, then Gand bG\G are separable and metizable, where P is a class of spaces which satisfy the followingconditions:(1) if X∈P, then every compact subset of the space X is a Gδ-set of X;(2) if X∈P and X is not locally compact, then X is not locally countably compact; (3) if X∈P and X is a Lindelo¨f p-space, then X is metrizable.Some known conclusions on rectifiable paracompact spaces and their remainders can begotten by this conclusion.In the last part, we mainly discuss a property of free topological groups and prove that ifthe topological space X belongs to P, then the free topological group generated by X belongsto σ-P, where P is a class of spaces which satisfy the following conditions:(1) if P∈P, then the image of P under a continuous map belongs to P;(2) if P1∈P and P2∈P, then P1∪P2∈P;(3) if P∈P and P2∈P, then P1×P2∈P.