Definition talk:Continuous Mapping (Topology)/Point

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The neighborhood $M$ of $x$ in Definition:Continuous Real Function at Point seems to be redundant. Any superset of a neighborhood of $x$ is just another neighborhood of $x$, so one might as well just specify that if $N$ is a neighborhood of $f \left({x}\right)$ in $T_2$, then $f^{-1} \left({N}\right)$ is a neighborhood of $x$ in $T_1$. Does anyone know if the sources: $\left({1}\right)$ use what I just mentioned, $\left({2}\right)$ use what is currently on the definition page, with the definition of a neighborhood as it is on this site, or $\left({3}\right)$ write "neighborhood" to mean "open neighborhood"? --abcxyz (talk) 02:40, 13 October 2012 (UTC)

This definition comes from 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): $\text{I}: \ \S 1$: Functions. They use "neighbourhood" to mean "not necessarily open neighbourhood". --prime mover (talk) 21:56, 3 November 2012 (UTC)
Hmm; I wonder why. --abcxyz (talk) 04:14, 4 November 2012 (UTC)
Also I find it in 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces where it is defined as open sets rather than neighborhoods. --prime mover (talk) 22:41, 3 November 2012 (UTC)