# Topology

## 1804 Submissions

[2] **viXra:1804.0234 [pdf]**
*replaced on 2018-04-19 22:40:24*

### An Algebraic Method for the Application of the Constructive Proof of Classification Theorem for Closed and Connected Surfaces

**Authors:** Onurcan Bektaş

**Comments:** 3 Pages.

For a given planar diagram of a closed & connected surface, we establish an ”algebraic” method for cutting and gluing operations on the edges of the diagram. By this, by just manipulating the name of the edges with the given rules, with the guidance of the classification of closed and connected surface theorem given in [1], we can determine the type of the surface without having need to draw any diagram.

**Category:** Topology

[1] **viXra:1804.0170 [pdf]**
*submitted on 2018-04-12 18:08:40*

### Retract Neutrosophic Crisp System for Gray Scale Image

**Authors:** A. A. Salama, Hewayda ElGhawalby, Asmaa.M.Nasr

**Comments:** 14 Pages.

In this paper, we aim to develop a new type of neutrosophic crisp set called the retract neutrosophic crisp set and shows a grayscale image in a 2D Cartesian domain with neutrosophic crisp components in the neutrosophic domain. The introduced set is a retraction of any triple structured crisp set. Whereas, the retractset deduced from any neutrosophic crisp set is coincide its corresponding star neutrosophic crisp set defined in by Salama et al. [1]. Hence we construct a new type of neutrosophic crisp topological spaces, called the retract neutrosophic crisp topological space as a retraction of the star neutrosophic topological space. Moreover, we investigate some of its properties.

**Category:** Topology