Embedding the free group 𝐹(𝑋) into 𝐹(𝛽𝑋)

Fay, Temple H.; Rajagopolan, M.; Smith-Thomas, Barbara V.

doi:10.1090/S0002-9939-1982-0637187-0

Embedding the free group $F(X)$ into $F(\beta X)$
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by Temple H. Fay, M. Rajagopolan and Barbara V. Smith-Thomas PDF

Proc. Amer. Math. Soc. 84 (1982), 297-302 Request permission

Abstract:

We show that for a Tychonoff space, $X$ and the canonical embedding ${\beta _X}:X \to \beta X$, the induced homomorphism $F{\beta _X}:F(X) \to F(\beta X)$ is an embedding between the free topological groups when $X$ has the property that ${X^n}$ is pseudo-compact for all $n \geqslant 1$. An application of this result is if $X$ is such a space and $\beta X$ is $0$-dimensional, then $F(X)$ is $0$-dimensional.

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