Rectifiable Spaces And Locally Minimal Paratopological Groups

This thesis is devoted to studying rectifiable spaces and locally Ti-minimal paratopological groups in the category of topological algebra. The contents are ar-ranged into four parts.In part1(Chapter2), we mainly discuss hereditarily normal rectifiable spaces and some cardinal invariants in rectifiable spaces. The main results are:(1) A heredi-tarily normal rectifiable space with a non-trivial convergent sequence has G△-diagonal (Theorem2.1.3);(2) Every compact subset of a hereditarily normal rectifiable space is metrizable (Theorem2.1.5);(3) Let G be a hereditarily normal rectifiable space. Then either G has a non-trivial convergent sequence and a Gδ-diagonal, or G has no non-trivial convergent sequences and every compact subset of G is finite. In either case, every compact subset of G is metrizable (Theorem2.1.6);(4) ω(G)≤ib(G)χ(G) is valid for each rectifiable space (Theorem2.2.2). Among them, the results (1),(2) and (3) generalize the corresponding results in [20] by Buzyakova.Part2(Chapter3) are arranged into two sections. In the first section we mainly discuss how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces itself and in the second section we discuss how the character of the remainders affect the character of rectifiable spaces itself. The main results are:(1) Let G be a non-locally compact rectifiable space with property (L1). If the remainder Y=bG\G has locally a property (L5), then G is separable metrizable and Y is a first-countable, Lindelof p-space (Theorem3.1.2);(2) Let G be a paracompact and non-locally compact rectifiable space, and Y=bG\G be locally symmetrizable. Then bG is separable and metrizable if each singleton of Y is a Gδ-set in Y (Theorem3.1.3);(3) Suppose that G is a non-locally compact rectifiable space with a remainder Y such that χ(Y)≤(?). Then χ(G)≤r+(Theorem3.2.1);(4) If G is a non-locally compact rectifiable space with a reminder Y satisfying χ(Y)≤(?), then|G|≤2(?)+(Theorem3.2.2);(5) Suppose that G is a non-locally compact recti-fiable space with a remainder Y such the tightness of Y is (?) and πχ(Y)≤(?)+.Then χ(G)<≤(?)+(Theorem3.2.3). These results improve the corresponding results of F.C. Lin, C. Liu [38] and A.V. Arhangel’skii, J. Van Mill [13]. In the third part (Chapter4), we discuss the locally property of connected and sequentially compact rectifiable spaces. The main results are:(1) If G is a lo-cally σ-sequentially compact rectifiable space with the Souslin property, then G is σ-sequentially compact (Theorem4.1.3);(2) Every connected locally a-sequentially compact rectifiable space G is σ-sequentially compact (Theorem4.1.6);(3) If every compact (resp. countably compact, sequentially compact) subspace of a rectifiable space G is Frechet-Urysohn, then every compact (resp. countably compact, sequen-tially compact) subspace of G is strongly Frechet-Urysohn (Theorem4.3.1).In the fourth part (Chapter5), we mainly discuss cardinal invariants in lo-cally Ti-minimal paratopological groups for i∈{1,2,3}. The main results are:(1) If (G, r) is a T2locally T1-minimal2-oscillating paratopological group, then X(G)=πχ(G)·inv(G)(Theorem5.2.1);(2) Let(G,(?)) be a locally T1-minimal paratopological group, then χ(G)=ψ(G)·inv(G)(Theorem5.2.2);(3) If (G,(?)) is a locally T2-minimal paratopological group, then χ(G)=ψ(G)·inv(G)·Hs(G)(Theorem5.3.1);(4) If (G,(?)) is a locally T3-minimal paratopological group, then X(G)=ψ[G)·inv(G)·Ir(G)(Theorem5.4.1). The corresponding results of F.C. Lin in [35] are improved respectively. We also proved the following two corollaries:(1) If G is a countable locally T3-minimal paratopological group, then G is metrizable (Corollary5.4.4);(2) If G is a locally T3-minimal paratopological group, then we have ω(G)=nω(G)(Corollary5.4.2). These two corollaries give positive answers to two questions posed by F.C. Lin in [35].