About Countable Dense Homogeneous spaces (CDH) and strongly locally homogeneous spaces

MathOverflow Asked by Gabriel Medina on January 3, 2022

I am new to the study of CDH topological spaces, I wanted to study basic examples of this type of spaces, for example I could understand the demonstration that $mathbb{R}$ is CDH, using the Cantor method (back-and- forth). I have tried to imitate the idea of proof to demonstrate that $mathbb{R}^{n}$ is CDH but I have not been successful, does anyone have any idea how to demonstrate this fact? I have found that other authors introduce the notion of Strongly Locally Homogeneous spaces, and show that "If $X$ is Polish and strongly locally
homogeneous. Then $X$ is countable dense homogeneous
". Unfortunately I have also failed to demonstrate that $mathbb{R}^{n}$ is strongly locally
homogeneous but also was not successful, does anyone have any ideas? Also, I would like to know what bibliography you recommend for a basic study of CDH spaces.

Thank you very much

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