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$G$-compactness for topological groups with operations

Osman Mucuk, Hüseyin Çakallı

Abstract

It is well known that for a Hausdorff topological group $X$, the limits of convergent sequences in $X$ define a function denoted by $\lim$ from the set of all convergent sequences in $X$ to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of $G$-continuity, $G$-compactness and $G$-connectedness. In this paper we prove some results on different types of $G$-compactness for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.

$G$-compactness for topological groups with operations

Abstract

It is well known that for a Hausdorff topological group , the limits of convergent sequences in define a function denoted by from the set of all convergent sequences in to . This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing with an arbitrary linear functional defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of -continuity, -compactness and -connectedness. In this paper we prove some results on different types of -compactness for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.

Paper Structure

This paper contains 4 sections, 20 theorems, 15 equations.

Table of Contents

  1. Preliminaries
  2. $G$-sequentially $\mathsf{T}_1$ topological groups with operations
  3. G-locally compactness of topological groups with operations
  4. Conclusion

Key Result

Proposition \oldthetheorem

For any subset $A$ of $X$ and a point $a\in A$, the subset $\{a\}$ is $G$-closed in $A$.

Theorems & Definitions (23)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 13 more