﻿ Erceg伪度量连续性公理及其基本球的关系的研究 The Researches of the Continuous Axiom of Erceg’s Pseudo-Metric and the Relationships between Its Basic Spheres

Vol.04 No.02(2015), Article ID:15351,7 pages
10.12677/AAM.2015.42026

The Researches of the Continuous Axiom of Erceg’s Pseudo-Metric and the Relationships between Its Basic Spheres

Peng Chen, Zhijuan Hu, Xiao Yang, Mengjie Jin, Leilei Liu, Zhigang Tian

Mathematics and Statistics Institute, Henan University of Science and Technology, Luoyang Henan

Email: chenpengbeijing@sina.com

Received: May 7th, 2015; accepted: May 22nd, 2015; published: May 29th, 2015

ABSTRACT

In this paper, by comparing the axioms of Erceg-Peng metric and classical metric, we have proved that there is no intrinsic relationship between the topology induced by Erceg metric and the continuous condition in its axioms, and further given some relationships of several types of basic spheres in Erceg-Peng’s pseudo-metric.

Keywords:Induced Topology, Erceg-Peng’s Axiom, Ur Open Mapping, Br Closed Mapping

Erceg伪度量连续性公理及其基本球的关系的研究

Email: chenpengbeijing@sina.com

1. 引言和预备

(A1)，如果，那么

(A2)

(A3)

(A4)，则

(B1)，如果，那么

(B2)

(B3)

(B4)使得使得

(B5)，如果

1)

2)

3)

4)

5)

3) 由。再根据引理1.5显然。

4) 由引理1.6得。

5) 从3)和4)有，因而(5)获得证明。

2. Fuzzy p-度量及与Erceg’s伪度量的关系

(D1)

(D2)

(D3)

(D4)

(D5)

1)是一个在上的拓扑基，记这个拓扑为

2)

3)

1)

2)

3)

3. Erceg伪度量函数的简化

(B3)*

.

(R1)

(R2)

(R3)

(R4)

(R2) 根据的定义，这能够被获得从定理3.4和(B2)。

(R3) 由。可证。

(R4) 根据(B4)和定理3.3易得(B4)。

,

(B1) 能够从(R1)获得。

(B2) 假设那么。因此。从(R2)知道，由此. 即

(B3) 由：可得。

(B4) 从(R2)和定理3.3可得。

1)

2)

2) 显然，让。从(B1)和(B2)，可得到。如果，那么有为每个，这暗示。因此。这是一个矛盾。因此，。从而

1)

2)

2) 如果，那么对每个。因此。从(1)，我们有。又因为。从引理1.7和定理3.7，我们可知

1)

2)

2) 显然。其次，如果，那么。因此我们有

The Researches of the Continuous Axiom of Erceg’s Pseudo-Metric and the Relationships between Its Basic Spheres. 应用数学进展,02,209-216. doi: 10.12677/AAM.2015.42026

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