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The definition of section is this:

In the mathematical field of topology, a section (or cross section) is a continuous right inverse of the projection function $\pi$. In other words, if $E$ is a fiber bundle over a base space, $B$:

$\pi : E \to B$

then a section of that fiber bundle is a continuous map,

$\sigma: B \to E$

such that

$\pi(\sigma(x)) = x$ for all $x \in B$.

Is there a name for the oject that are defined so, that we omit the continuity from the definition of the section? In other words, I am looking for a correct (standard) replacement word for (what) here:

In the mathematical field of topology, a (what) is a right inverse of the projection function $\pi$. In other words, if $E$ is a fiber bundle over a base space, $B$:

$\pi : E \to B$

then a (what) of that fiber bundle is a map,

$\sigma: B \to E$

such that

$\pi(\sigma(x)) = x$ for all $x \in B$.

Motivation: I'd like to find a standard name for the set of the points in the 3-dimensional space that correspond to the points of a given photo. Here the fibers are the 1-dimensional subspaces (or cones) of the physical space, and the base space is the photo itself. In general, it isn't a section because of the possible lack of continuity.

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    $\begingroup$ It's just a right inverse. This works in any category. A continuous right inverse is a section, as you say. A selector for $\pi$ would be a reasonable name too, as we "select" a point in each fibre. $\endgroup$ – Henno Brandsma Aug 12 '17 at 6:17
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Lee (in “Introduction to Smooth Manifolds“) calls these maps “rough sections“ to stress the difference from continuous or even smooth sections.

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You could call it a "right inverse to the projection", a "set-theoretic section", or a "possibly discontinuous section". If you want a brief term, I would recommend simply saying "section" and mentioning once that you do not assume sections are continuous. Indeed, it is not really universally standard to assume that sections are continuous; rather, "section" is used as a shorthand for "continuous section" much the same way "map" is used as a shorthand for "continuous map".

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    $\begingroup$ Also the phrase "not-necessarily continuous section" conveys the right idea. $\endgroup$ – goblin Aug 12 '17 at 6:22
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These are called any/all of:

  • Set-theoretic sections
  • Sections in the category of sets
  • Sections in $\mathbf{Set}$
  • Sections with respect to the category structure on the groupoid of topological manifold in which morphisms are functions.
  • etc.

See here for more information.

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I already heard the word lift or lifting for this, similarly to this

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