Corollary 1

Tag 04E4, permanent. Cite it as [Bourbaki, Tag 04E4].

Suppose that X is Hausdorff and its filtration exhaustive. Then, if gr(u) is injective, u is injective.

Let e, e' be the identity elements of X and Y respectively. Then

\bar{u}^{-1}(e') \subset \bigcap_n \bar{u}^{-1}(Y_n) = \bigcap_n X_n = \{e\}

by hypothesis, whence the corollary.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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