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F{\display style f} being topologically semi conjugate to g{\display style g} means, by definition, that h{\display style h} is a subjection such that FSH=hog{\display style f\CIRC h=h\CIRC g}. For certain values in the parameter space, the Henson map when restricted to its Julia set is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.

In fact, we are requiring the maps (, t){\display style \var phi (\dot, t)} and (, t){\display style \psi (\dot, t)} to be topologically conjugate for each t{\display style t}, which is requiring more than simply that orbits of {\display style \phi} be mapped to orbits of {\display style \psi} monomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in X{\display style X} into classes of flows sharing the same dynamics, again from the topological viewpoint.

Overall, topological equivalence is a weaker equivalence criterion than topological conjugate, as it does not require that the time term is mapped along with the orbits and their orientation. An example of a topologically equivalent but not topologically conjugate system would be the non-hyperbolic class of two-dimensional systems of differential equations that have closed orbits.

While the orbits can be transformed to each other to overlap in the spatial sense, the periods of such systems cannot be analogously matched, thus failing to satisfy the topological conjugate criterion while satisfying the topological equivalence criterion. More equivalence criteria can be studied if the flows, {\display style \phi} and {\display style \psi}, arise from differential equations.

Two dynamical systems defined by the differential equations, X=f(x){\display style {\dot {x}}=f(x)} and yo=g(y){\display style {\dot {y}}=g(y)}, are said to be smoothly equivalent if there is a diffeomorphism, h:XY{\display style h:X\to Y}, such that Two dynamical systems on the same state space, defined by X=f(x){\display style {\dot {x}}=f(x)} and X=g(x){\display style {\dot {x}}=g(x)}, are said to be orbitally equivalent if there is a positive function, :XR{\display style \mu :X\to \math {R}}, such that g(x)=(x)f(x){\display style g(x)=\mu (x)f(x)}.

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Orbitally equivalent system differ only in the time parametrization. For example, consider linear systems in two dimensions of the form X=Ax{\display style {\dot {x}}=Ax}.

If the matrix, A{\display style A}, has two positive real eigenvalues, the system has an unstable node; if the matrix has two complex eigenvalues with positive real part, the system has an unstable focus (or spiral). Nodes and loci are topologically equivalent but not orbitally equivalent or smoothly equivalent, because their eigenvalues are different (notice that the Jacobi ans of two locally smoothly equivalent systems must be similar, so their eigenvalues, as well as algebraic and geometric multiplicities, must be equal).

There are two reported extensions of the concept of dynamic topological conjugate: Analogous systems defined as isomorphic dynamical systems Adjoint dynamical systems defined via adjoint functors and natural equivalences in categorical dynamics.

The Volcano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval , some of those points will get arbitrarily close to some real number in that space.

In particular, the sequence of points 0, 1, 2, 3, …, which is not bounded, has no subsequence that converges to any real number. The concept of a compact space was formally introduced by Maurice Freshet in 1906 to generalize the Volcano–Weierstrass theorem to spaces of functions, rather than geometrical points.

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In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that cover the space in the sense that each point of the space lies in some set contained in the family.

In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. Volcano's proof relied on the method of bisection : the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected.

The full significance of Volcano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass. In the 1880s, it became clear that results similar to the Volcano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points.

The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio ASCII and Cesare Area. Towards the beginning of the twentieth century, results similar to that of Area and ASCII began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt.

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.

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The significance of this lemma was recognized by Émile Bore (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Heine–Bore theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.

This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearing his name.

Ultimately, the Russian school of point-set topology, under the direction of Pavel Alexandros and Pavel Bryson, formulated Heine–Bore compactness in a way that could be applied to the modern notion of a topological space. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

A compact set is sometimes referred to as a compact um, plural compact. X has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (Alexander's sub-base theorem) X is Lancelot and countably compact Any collection of closed subsets of X with the finite intersection property has nonempty intersection.

(X, d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces). Lebesgue's number lemma : For every open cover of X, there exists a number > 0 such that every subset of X of diameter < is contained in some member of the cover.

The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. A topological SpaceX is pseudocompact if and only if every maximal ideal in C (X) has residue field the real numbers.

For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism. The framework of non-standard analysis allows for the following alternative characterization of compactness: a topological SpaceX is compact if and only if every point x of the natural extension *X is infinitely close to a point x 0 of X (more precisely, x is contained in the monad of x 0).

Hyperreal definition A SpaceX is compact if its hyperreal extension *X (constructed, for example, by the ultrapower construction) has the property that every point of *X is infinitely close to some point of X *X. For example, an open real interval X = (0, 1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X.

In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed.

(Slightly more generally, this is true for an upper semi continuous function.) Any finite topological space, including the empty set, is compact.

Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandros one-point compactification. No discrete space with an infinite number of points is compact.

The collection of all singletons of the space is an open cover which admits no finite subcover. In carrying the lower limit topology, no uncountable set is compact.

Like the previous example, the space as a whole is not locally compact but is still Lancelot. Similarly, the set of rational numbers in the closed interval is not compact: the sets of rational numbers in the intervals and {\display style \left\ {\text{and}}\ \left} cover all the rationals in for n = 4, 5, ... but this cover does not have a finite subcover.

The set of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals (n 1, n +1), where n takes all integer values in Z, cover but there is no finite subcover.

On the other hand, the extended real number line carrying the analogous topology is compact; note that the cover described above would never reach the points at infinity. In fact, the set has the homomorphism to of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homomorphism preserve covers, the Heine-Borel property can be inferred.

Again from the Heine–Bore theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. In fact, every compact metric space is a continuous image of the Cantor set.

Consider the set K of all functions f : from the real number line to the closed unit interval, and define a topology on K so that a sequence {f n} {\display style \{f_{n}\}} in K converges towards f K if and only if {f n (x) } {\display style \{f_{n}(x)\}} converges towards f (x) for all real numbers x. The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers .

Conversely, any compact subset of arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space 2 {\display style \ell ^{2}} may have any compact nonempty subset of as spectrum.

The spectrum of any commutative ring with the Paris topology (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases). The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem.

The Hilbert cube is compact, again a consequence of Tychonoff's theorem. We endow X with the particular point topology by defining a subset U X to be open if and only if 0 U.

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