''of a function, set of values of a function, image of a function''
...$x \in X$ with $f(x) = y$. Thus, the range of values of a function is the image of its [[domain of definition]], $Y_f = f(X)$.
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...[zero-dimensional space]]. Also, every [[compact space]] is the continuous image of a closed subspace of a Cantor cube.
The terminology is slightly ambiguous: a [[dyadic space]] is a continuous image of a Cantor cube, so that "dyadic discontinuum" could also mean "totally
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...the area $S(E)$ of that part of the convex surface $F$ that has spherical image $E\subset\Omega$. This definition remains meaningful for general convex sur
...lled the spherical image of $x$. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. [[Spherical map|Sp
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$#C+1 = 33 : ~/encyclopedia/old_files/data/I050/I.0500150 Image of a morphism
A concept similar to that of the image of a mapping of one set into another. However, in category theory there are
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$#C+1 = 12 : ~/encyclopedia/old_files/data/S085/S.0805800 Small image
is closed (cf. [[Closed mapping|Closed mapping]]) if and only if the small image $ f ^ { \sharp } U $
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...ts with $n=1$. After the introduction of a suitable topology, the analytic image is converted to the [[Riemann surface]] of the given function.
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...us mapping|Continuous mapping]]) of topological spaces under which the pre-image of every point is compact. Perfect mappings are akin to continuous mappings
...pre-image, and the Suslin number of the image is equal to that of the pre-image. If a completely-regular $T_1$-space is mapped onto a completely-regular $T
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...ssible to select a finite number of sets so that $y$ is located inside the image of their union. It is particularly important that the product of any collec
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The topological image of a [[Disc|disc]] (cf. also [[Disc, topological|Disc, topological]]).
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A homeomorphic image of a circle (cf. also [[Homeomorphism|Homeomorphism]]). Named after C. Jord
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A continuous image of a [[Borel set]]. Since any Borel set is a continuous
image of the set of irrational numbers, an ${\mathcal A}$-set can be defined as a
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is entirely covered by the image of the disc $ | z | < 1 $
there are points not belonging to the image only if $ f $
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...space]] such that $\pi(x)^*=\pi(x^*)$ for all $x\in A$, where $x^*$ is the image of $x$ under the involution of $A$.
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...w Y$ is a morphism of algebraic varieties, then $f(X)$ (and, moreover, the image under $f$ of any constructible subset in $X$) is a constructible subset in
...ed constructible if $h(X)$ is finite and if for any point $t\in T$ the pre-image $h^{-1}(t)$ is a constructible subset in $X$.
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...onto a topological space $Y$ which is irreducible (that is, $Y$ is not the image of any proper closed subset of $X$, cf. also [[Irreducible mapping|Irreduci
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...xtensions $A \subset \Omega$ and $B \subset \Omega$. It is the same as the image of the homomorphism $ \phi : A \otimes_{k} B \to \Omega$ that maps the tens
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...s such particularly useful for the analysis of geometrical structure in an image. It has been applied in various disciplines, such as mineralogy, medical di
...and their interrelations. In general, this procedure results in non-linear image operators, some of which will be described briefly below.
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A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mapping
...e the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in
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...delta$-diagonal are also metrizable. The perfect image and the perfect pre-image of a paracompact $P$-space are also paracompact $P$-spaces.
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...puter vision, aimed at detecting and characterizing discontinuities in the image domain.
...on in terms of edges is also compact in the sense that the two-dimensional image pattern is represented by a set of one-dimensional curves. For these reason
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...all inverse images of points are finite. Mappings under which the inverse image of every compact set is compact are called $ k $-
is open if and only if its complete inverse image $ f ^ { - 1 } ( V) $
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A mapping of one topological space into another under which the image of every open set is itself open.
...by a complete metric. If a [[Paracompact space|paracompact space]] is the image of a [[Complete metric space|complete metric space]] under a continuous ope
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...f the $2$-dimensional sphere $S^2$; also, every cactoid is a monotone open image of $S^2$.
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...tremal disconnectedness is not preserved by perfect mappings. However, the image of an extremally-disconnected space under a continuous open mapping is an e
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...et: \(X\) satisfies the first axiom of countability; \(X\) is a continuous image of an ordered compactum; \(X\) is hereditarily normal; \(X\) is hereditaril
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A '''Suslin space''' is a continuous image of a Polish space.
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...rplane section in $\PP^n$. The Veronese mapping is a closed imbedding; its image $v_m(\PP^n)$ is called a Veronese variety, and is defined by the equations
in $\PP^n$ its image with respect to the Veronese mapping $v_m$ is the intersection of the Veron
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a pre-image $ x \in L _ {n} $
will then be the image of an element $ z \in Z _ {n-} 1 ( K _ {\mathbf . } ) $
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..., depending on the orientation. Certain knots can differ from their mirror image owing to the fact that their branched coverings are asymmetric varieties.
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...nding to the deformation over a principal base, as well as to the Clifford image of the asymptotic lines of a surface in an elliptic space (which is the [[R
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It follows from the Tarski–Seidenberg theorem that the image under a polynomial mapping $ \mathbf R ^ {n} \rightarrow \mathbf R ^ {m
...t on a real-analytic manifold is, by definition, a set that is locally the image of a semi-analytic set under an analytic mapping. The points of a subanalyt
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...ts of the kernel $\ker d$ are known as ''cycles''; and the elements of the image $\mathrm{im}\, d$ are called ''boundaries''. The ''homology'' of $C$ is th
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...rithmic capacity|logarithmic capacity]]. As an example, any function whose image does not contain a disc of a fixed radius $\epsilon > 0$ centred at $a +
...proved that an analytic function is in $\operatorname{BMOA}$ provided its image [[Riemann surface]] (viewed as spread out over the complex plane) has the f
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The Lindelöf principle states that under these conditions: 1) the inverse image $ \widetilde{D} _ \rho $
lies inside the inverse image $ D _ \rho $
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...an open connected set. The basic property here is that of openness of the image, which follows from Rouché's theorem or from the principle of the argument
say, then the image $ f ( X) \subset Y $
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..., touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. band is the Zhukovskii profile.
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A continuous mapping $h : f(X) \rightarrow X$ of the image $f(X)$ of a continuous mapping $f : X \rightarrow Y$ of topological spaces
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is always the image of some horizontal distribution $ \Delta $
is obtained as the image $ \pi ^ {*} \overline \Delta \; $,
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such that the image of any set $ U $
while the inverse image $ f ^ { - 1 } ( y) $
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...z$ vary monotonically on the image of the trajectory in $|z|<\alpha$. Each image of a trajectory twists around the point $z=0$ and behaves asymptotically li
...ha$ in the other direction. The modulus of $z$ varies monotonically on the image of the trajectory in $|z|<\alpha$. Different images of trajectories have di
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...ontinuum. The topological product of continua is a continuum, a continuous image of a continuum is a continuum, the components of a Hausdorff compactum are
Every locally connected metric continuum is a continuous image of a closed segment (the Hahn–Mazurkiewicz theorem). A non-degenerate con
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...continuous mapping $g : X \to Q^n$ that coincides with $f$ on the inverse image $f^{-1} S^{n-1}$ of the boundary $S^{n-1}$ of $Q^n$ and takes $X$ into $S^{
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...ath.org/legacyimages/c/c025/c025720/c02572018.png" /> the complete inverse
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...ediaofmath.org/legacyimages/c/c025/c025720/c02572049.png" />, that is, the
image of any bounded set <img align="absmiddle" border="0" src="https://www.encyc
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...orithms, aimed at detecting and classifying the nature of junctions in the image domain. A main reason why corner detection is important is that junctions p
One way of detecting junctions directly from image intensities consists of finding points at which the gradient magnitude $| \
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...try with respect to a circle). The centre of an inversion does not have an image. An inversion with negative power $k$ is equivalent to the inversion with t
Sometimes an [[ideal point]] $\infty$ is regarded as the image of the centre of an inversion under this inversion, especially when one con
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...mbedding]]. If a curve $X$ is not hyper-elliptic and has genus 2, then its image in the projective space $P^{g-1}$ under a canonical imbedding has degree $2
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...$u$ of a set $X$ (in general endowed with some structure) into itself. The image of an element $x \in X$ under the transformation $u$ is denoted by $u(x)$,
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...encyclopediaofmath.org/legacyimages/c/c025/c025710/c0257105.png" /> of its
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
a) the inverse
image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
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...eta$ if $\alpha$ is a limit ordinal number; $Z_\alpha$ is the complete pre-image of the centre of the quotient group $G/Z_\beta$ if $\alpha = \beta+1$ is a
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...f $S$ is a normal sub-semi-group if and only if it is the complete inverse image of the [[unit element]] under some homomorphism of $S$ onto a [[Monoid|semi
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