# Cartan's theorem for some topological generalized groups

## TL;DR

This paper extends Cartan's theorem to certain topological generalized groups, establishing conditions under which topological subgroupoids are Lie subgroupoids and exploring their structural properties.

## Contribution

It introduces conditions that ensure topological subgroupoids of Lie groupoids are Lie subgroupoids and generalizes Cartan's theorem to generalized Lie groups.

## Key findings

- Topological subgroupoids are Lie subgroupoids under specific conditions
- Same topological dimension is necessary for subgroupoids to be Lie subgroupoids
- Conditions for double subgroupoids to become double Lie subgroupoids

## Abstract

In this paper we show that topological subgroupoids of Lie groupoids, under special circumstances are Lie subgroupoids. Giving an example, we indicate that having the same topological dimension is a necessary condition for topological subgroupoids to be Lie subgroupoids. Also, we provide some conditions for double subgroupoids to become double Lie subgroupoids. Moreover, we illustrate that having the same conditions as the Cartan's theorem for Lie groups, helps us prove the same theorem for generalized Lie groups.