Definition:Asymmetric Relation/Also defined as

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Asymmetric Relation Also defined as

Some sources (possibly erroneously or carelessly) gloss over the differences between this and the definition for an antisymmetric relation, and end up using a definition for antisymmetric which comes too close to one for asymmetric.

An example is 1964: Steven A. Gaal: Point Set Topology:

[After having discussed antireflexivity] ... antisymmetry expresses the additional fact that at most one of the possibilities $a \mathrel \RR b$ or $b \mathrel \RR a$ can take place.

Some sources specifically define a relation as anti-symmetric what has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as asymmetric

From 1955: John L. Kelley: General Topology: Chapter $0$: Relations:

... the relation $R$ is anti-symmetric iff it is never the case that both $x R y$ and $y R x$.