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The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
Some authors call a function : between two topological spaces proper if the preimage of every compact set in is compact in . Other authors call a map f {\displaystyle f} proper if it is continuous and closed with compact fibers ; that is if it is a continuous closed map and the preimage of every point in Y {\displaystyle Y} is compact .
If and are the domain and image of , respectively, then the fibers of are the sets in {():} = {{: =}:}which is a partition of the domain set .Note that must be restricted to the image set of , since otherwise () would be the empty set which is not allowed in a partition.
If A is a Lebesgue-measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set. A fortiori , every subset of A is measurable. If A is Lebesgue-measurable and x is an element of R n , then the translation of A by x , defined by A + x = { a + x : a ∈ A }, is also Lebesgue-measurable and has the same measure as A .
The standard construction of the Cantor set is an example of a null uncountable set in ; however other constructions are possible which assign the Cantor set any measure whatsoever. All the subsets of R n {\displaystyle \mathbb {R} ^{n}} whose dimension is smaller than n {\displaystyle n} have null Lebesgue measure in R n . {\displaystyle ...
The indicator or characteristic function of a subset A of some set X maps elements of X to the codomain {,}. This mapping is surjective only when A is a non-empty proper subset of X . If A = X , {\displaystyle A=X,} then 1 A ≡ 1. {\displaystyle \mathbf {1} _{A}\equiv 1.}
Conversely, if is a Hausdorff space and is a closed set, then the coimage of , if given the quotient space topology, must also be a Hausdorff space. A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty; [ 4 ] [ 5 ] said differently, a space is compact if ...
YouTube Rewind 2010: Year in Review and YouTube Rewind 2011, however, have less than 10 million views each. The Ultimate 2016 Challenge became YouTube's fastest video to reach 100 million views, doing so in just 3.2 days. It is also the eighth most-liked non-music video of all time with over 3.40 million likes.