Hubbry Logo
Isolated pointIsolated pointMain
Open search
Isolated point
Community hub
Isolated point
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Isolated point
Isolated point
Isolated point
View on Wikipedia
from Wikipedia
"0" is an isolated point of

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S.

If the space X is a metric space, for example a Euclidean space, then an element x of S is an isolated point of S if there exists an open ball around x that contains only finitely many elements of S. A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).

[edit]

Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of S may be mapped injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.

A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it contains all its limit points and no isolated points).

The number of isolated points is a topological invariant, i.e. if two topological spaces X, Y are homeomorphic, the number of isolated points in each is equal.

Examples

[edit]

Standard examples

[edit]

Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.

  • For the set the point 0 is an isolated point.
  • For the set each of the points ⁠⁠ is an isolated point, but 0 is not an isolated point because there are other points in S as close to 0 as desired.
  • The set of natural numbers is a discrete set.

In the topological space with topology the element a is an isolated point, even though belongs to the closure of (and is therefore, in some sense, "close" to a). Such a situation is not possible in a Hausdorff space.

The Morse lemma states that non-degenerate critical points of certain functions are isolated.

Two counter-intuitive examples

[edit]

Consider the set F of points x in the real interval (0,1) such that every digit xi of their binary representation fulfills the following conditions:

  • Either or
  • only for finitely many indices i.
  • If m denotes the largest index such that then
  • If and then exactly one of the following two conditions holds: or

Informally, these conditions means that every digit of the binary representation of that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.

Now, F is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.[1]

Another set F with the same properties can be obtained as follows. Let C be the middle-thirds Cantor set, let be the component intervals of , and let F be a set consisting of one point from each Ik. Since each Ik contains only one point from F, every point of F is an isolated point. However, if p is any point in the Cantor set, then every neighborhood of p contains at least one Ik, and hence at least one point of F. It follows that each point of the Cantor set lies in the closure of F, and therefore F has uncountable closure.

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Isolated point

View on Grokipedia
from Grokipedia
In , particularly in the field of , an isolated point of a subset SS of a XX is a point x∈Sx \in S such that there exists an open neighborhood UU of xx with U∩S={x}U \cap S = \{ x \}. This means xx is separated from all other points in SS by some open set containing only xx from SS. Equivalently, xx is an isolated point if it is not a limit point of SS, where a limit point requires every open neighborhood of the point to contain at least one other point from SS. Isolated points play a key role in characterizing the structure of subsets in topological spaces, distinguishing them from accumulation or limit points that form clusters. In metric spaces like the real line R\mathbb{R}, a point x∈A⊂Rx \in A \subset \mathbb{R} is isolated if there exists δ>0\delta > 0 such that (x−δ,x+δ)∩A={x}(x - \delta, x + \delta) \cap A = \{ x \}. For example, every natural number in the set N⊂R\mathbb{N} \subset \mathbb{R} is an isolated point, as each has a neighborhood containing no other naturals, making N\mathbb{N} a discrete subset with no limit points. Similarly, in the set A={1/n:n∈N}⊂RA = \{ 1/n : n \in \mathbb{N} \} \subset \mathbb{R}, each 1/n1/n is isolated, though adding 0 would make 0 a limit point while the others remain isolated. The absence of isolated points defines perfect sets, which are closed subsets equal to their own derived set (the set of all limit points), such as the in R\mathbb{R}. In the discrete topology on any set, every point is isolated, as singletons are open. Isolated points contribute to the closure of a set, where the closure is the union of the set and its limit points, excluding isolated points only if they are already included. This concept extends to more abstract spaces, aiding in the study of continuity, , and connectedness.

Definition

Formal definition

A topological space is a pair (X,τ)(X, \tau) consisting of a set XX and a collection τ\tau of subsets of XX, called open sets, that satisfies the following axioms: the ∅\emptyset and XX itself are in τ\tau; the arbitrary union of any collection of sets in τ\tau is in τ\tau; and the finite intersection of any collection of sets in τ\tau is in τ\tau. Let (X,τ)(X, \tau) be a and S⊆XS \subseteq X a . A point x∈Sx \in S is called an isolated point of SS if there exists an open neighborhood U∈τU \in \tau of xx (that is, x∈Ux \in U) such that U∩S={x}U \cap S = \{x\}./Subset/Definition_1) This intersection condition ensures that xx is separated from all other points of SS within some open set containing it. The notion of an isolated point arose in the development of point-set topology in the late 19th and early 20th centuries, with foundational contributions from in his 1914 monograph Grundzüge der Mengenlehre, which systematized concepts like neighborhoods and separation in abstract spaces. In opposition to isolated points, limit points of SS are those where every open neighborhood intersects SS in points other than itself.

Equivalent characterizations

In a topological space XX, a point x∈S⊆Xx \in S \subseteq X is an isolated point of SS if and only if the singleton {x}\{x\} is an open set in the subspace topology induced on SS. This equivalence holds because the subspace topology consists of sets of the form U∩SU \cap S where UU is open in XX, so {x}\{x\} being open in SS means there exists an open neighborhood UU of xx in XX such that U∩S={x}U \cap S = \{x\}. Equivalently, xx is an isolated point of SS if and only if xx is not a limit point of SS, meaning x∉S′x \notin S', where S′S' denotes the derived set of limit points of SS. To see this, suppose xx is isolated via the neighborhood condition; then the open set UU with U∩S={x}U \cap S = \{x\} contains no other points of SS, so no neighborhood of xx intersects S∖{x}S \setminus \{x\} nontrivially, implying xx is not a limit point. Conversely, if xx is not a limit point, then there exists an open neighborhood VV of xx such that V∩(S∖{x})=∅V \cap (S \setminus \{x\}) = \emptyset, so V∩S={x}V \cap S = \{x\}, satisfying the isolation condition. This characterization extends to the closure operator, as xx is isolated in SS if and only if x∉cl(S∖{x})x \notin \mathrm{cl}(S \setminus \{x\}), where cl(A)\mathrm{cl}(A) denotes the closure of a set A⊆XA \subseteq X. Indeed, the definition of limit point states that x∈S′x \in S' precisely when x∈cl(S∖{x})x \in \mathrm{cl}(S \setminus \{x\}), since every open neighborhood of xx intersects S∖{x}S \setminus \{x\} if and only if it intersects the complement of {x}\{x\} in SS. Using the relation cl(A)=A∪A′\mathrm{cl}(A) = A \cup A' for any A⊆XA \subseteq X, the condition x∉cl(S∖{x})x \notin \mathrm{cl}(S \setminus \{x\}) directly implies x∉(S∖{x})′x \notin (S \setminus \{x\})', confirming xx is not a limit point of SS. From the perspective of the subspace topology on SS, xx is isolated if the relative open sets around xx form a neighborhood basis consisting solely of the singleton {x}\{x\}. This follows immediately from {x}\{x\} being open in SS, as it serves as a local basis element at xx in the relative topology. The set of all isolated points of SS comprises the discrete component S∖S′S \setminus S', allowing SS to be partitioned into its isolated points and its limit points. This decomposition highlights that isolated points form a discrete subset of SS, separated from the accumulation structure captured by the derived set S′S'.

Properties

In topological spaces

In a topological space XX, for a subset S⊆XS \subseteq X, the set of isolated points of SS, denoted I(S)I(S), is the complement of the derived set S′S' in SS, where the derived set S′S' consists of all limit points of SS. Thus, I(S)=S∖S′I(S) = S \setminus S'. A point x∈Sx \in S is isolated in SS if and only if the singleton {x}\{x\} is open in the subspace topology on SS. The collection of all such singletons for isolated points in SS therefore forms an open subset of SS in the subspace topology, as it is a union of open sets. If SS has only finitely many isolated points, this open subset coincides exactly with the finite set of those points. Homeomorphisms preserve isolated points: if f:X→Yf: X \to Y is a homeomorphism and x∈S⊆Xx \in S \subseteq X is isolated in SS, then {x}\{x\} open in SS implies f({x})={f(x)}f(\{x\}) = \{f(x)\} open in f(S)f(S), so f(x)f(x) is isolated in f(S)f(S). This follows from homeomorphisms mapping open sets to open sets and being bijective. In a second-countable topological space XX, the set of isolated points of any subset S⊆XS \subseteq X is at most countable. Each isolated point in SS admits a distinct basis element from the countable basis of XX that intersects SS only at that point, implying at most countably many such points.

In metric spaces

In a (X,d)(X, d), a point x∈S⊆Xx \in S \subseteq X is isolated in SS if there exists ε>0\varepsilon > 0 such that the open ball B(x,ε)={y∈X∣d(x,y)<ε}B(x, \varepsilon) = \{ y \in X \mid d(x, y) < \varepsilon \} satisfies B(x,ε)∩S={x}B(x, \varepsilon) \cap S = \{x\}. This condition ensures that xx is separated from the rest of SS by a positive distance within the metric structure. The isolation can be quantified by the isolation radius of xx in SS, defined as r(x)=inf⁡{d(x,y)∣y∈S,y≠x}r(x) = \inf \{ d(x, y) \mid y \in S, y \neq x \}. For xx to be isolated, r(x)>0r(x) > 0. This radius provides a measure of separation, equivalent to the existence of the open ball condition in metric spaces, as metrics induce structures. Specifically, xx is isolated in SS the from xx to S∖{x}S \setminus \{x\} is positive, i.e., d(x,S∖{x})>0d(x, S \setminus \{x\}) > 0. To see the equivalence, suppose xx is isolated with corresponding ε>0\varepsilon > 0. Then for any y∈S∖{x}y \in S \setminus \{x\}, y∉B(x,ε)y \notin B(x, \varepsilon), so d(x,y)≥ε>0d(x, y) \geq \varepsilon > 0, implying r(x)≥ε>0r(x) \geq \varepsilon > 0. Conversely, if r(x)=δ>0r(x) = \delta > 0, take ε=δ\varepsilon = \delta. If there exists z∈B(x,ε)∩Sz \in B(x, \varepsilon) \cap S with z≠xz \neq x, then d(x,z)<δ≤d(x,z)d(x, z) < \delta \leq d(x, z), a contradiction. Thus, B(x,ε)∩S={x}B(x, \varepsilon) \cap S = \{x\}. This characterization relies directly on the metric properties without needing the triangle inequality for the isolation itself, though the triangle inequality underpins the ball's role in defining proximity. A key implication for convergence is that if xx is isolated in SS, no sequence of distinct points from SS can converge to xx. Any sequence {xn}⊆S\{x_n\} \subseteq S converging to xx must be eventually constant, equal to xx for all sufficiently large nn. To see this, suppose {xn}\{x_n\} converges to xx but xn≠xx_n \neq x for infinitely many nn. Then, for the ε>0\varepsilon > 0 from the isolation condition, infinitely many xnx_n lie in B(x,ε)∩S∖{x}=∅B(x, \varepsilon) \cap S \setminus \{x\} = \emptyset, impossible. In complete metric spaces, closed subsets inherit completeness, and isolated points of such subsets maintain their isolation relative to the subset via the ambient metric. If SS is a closed of a complete metric space XX, and x∈Sx \in S is isolated in SS with radius r(x)>0r(x) > 0, the same open ball B(x,r(x))B(x, r(x)) intersects SS only at xx, preserving the property without alteration from the completeness of XX or SS. This ensures that isolation, being a local metric feature, is robust under closure operations in complete settings.

Examples

Standard examples

In the real line R\mathbb{R} equipped with the standard topology, the subspace consisting of the integers Z\mathbb{Z} provides a classic example where every point is isolated. For each integer n∈Zn \in \mathbb{Z}, the open interval (n−1/2,n+1/2)(n - 1/2, n + 1/2) is an open neighborhood of nn that intersects Z\mathbb{Z} solely at nn, satisfying the isolation condition. Finite subsets of R\mathbb{R} also exhibit isolated points exclusively. In any finite set A={a1,…,ak}⊂RA = \{a_1, \dots, a_k\} \subset \mathbb{R} with the subspace topology, each aia_i is isolated because the minimum distance d=min⁡i≠j∣ai−aj∣/2>0d = \min_{i \neq j} |a_i - a_j|/2 > 0 allows an open ball of radius dd around aia_i to contain no other points of AA. This holds more generally in any , where finite subsets have all points isolated. Consider the subspace A={1/n:n∈N}⊂RA = \{1/n : n \in \mathbb{N}\} \subset \mathbb{R}. Every point 1/n∈A1/n \in A is isolated, as a sufficiently small open interval around 1/n1/n—smaller than the distance to the nearest other point in AA—intersects AA only at 1/n1/n. In contrast, the subspace of rational numbers Q⊂R\mathbb{Q} \subset \mathbb{R} has no isolated points, since every open interval around any rational contains infinitely many other rationals. Extending this, the set {0}∪{1/n:n∈N}\{0\} \cup \{1/n : n \in \mathbb{N}\} has all 1/n1/n as isolated points, while 00 is a limit point. In any set equipped with the discrete topology, where every subset is open, all points are isolated by definition, as the singleton {x}\{x\} serves as an open neighborhood containing only xx.

Counterexamples

In the , constructed as the intersection of a nested sequence of closed intervals in [0,1][0,1] by iteratively removing middle thirds, every point serves as a limit point despite the set's "dust-like" appearance, which might intuitively suggest isolated components; thus, the set contains no isolated points and is dense-in-itself. The set of rational numbers Q\mathbb{Q} embedded in R\mathbb{R} with the standard topology provides another , as its density ensures that every rational is a limit point approachable by other rationals arbitrarily closely, contradicting the intuition of "gaps" between rationals that might imply isolation; consequently, Q\mathbb{Q} has no isolated points. Consider the rational sequence topology on R\mathbb{R}, where a basis consists of singleton sets for each rational and, for each irrational xx with a sequence of rationals (qn)(q_n) converging to xx in the standard topology, neighborhoods Un(x)={qk∣k≥n}∪{x}U_n(x) = \{q_k \mid k \geq n\} \cup \{x\}; here, the rationals become isolated points, but the irrationals do not, as each is a limit point of its approximating rational , illustrating how modifying the on R∖Q\mathbb{R} \setminus \mathbb{Q} via added sequences prevents overall isolation while creating a non-metrizable space. In the indiscrete topology on a XX with more than one point, the only open sets are ∅\emptyset and XX itself, so no singleton {x}\{x\} is open, rendering every point non-isolated even though singletons might seem trivially separated; this extreme coarseness makes all points limit points of any non-trivial , serving as a to expectations in non-Hausdorff settings. The Sorgenfrey line, defined on R\mathbb{R} with basis elements [a,b)[a, b) for a<ba < b, yields no isolated points overall, as every basis neighborhood [x,x+ϵ)[x, x + \epsilon) of a point xx contains uncountably many other points to the right; although points appear "isolated from the left," the dense structure ensures limit points abound, countering partial intuitions of rightward isolation, particularly in dense subsets like the irrationals.

Applications

In analysis

In , the zeros of a non-constant are isolated points. Specifically, if ff is analytic in a domain D⊂CD \subset \mathbb{C} and f(z0)=0f(z_0) = 0 for some z0∈Dz_0 \in D, then there exists a neighborhood around z0z_0 containing no other zeros of ff. The identity theorem extends this by stating that if the set of zeros of ff in a connected has a limit point in that set, then ff is identically zero throughout the set. Isolated singularities play a key role in classifying points where analytic functions fail to be holomorphic. A singularity at z0z_0 is removable if the function can be redefined at z0z_0 to become analytic there, which occurs precisely when lim⁡z→z0(z−z0)f(z)=0\lim_{z \to z_0} (z - z_0) f(z) = 0. Riemann's removable singularity theorem guarantees that if ff is analytic and bounded in a punctured disk around an isolated singularity z0z_0, then the singularity is removable, allowing an analytic extension to the full disk. In on subsets of R\mathbb{R}, isolated points affect the convergence of . If pp is an isolated point of a S⊆RS \subseteq \mathbb{R}, then any in SS converging to pp must be the constant sequence p,p,p,…p, p, p, \dots, as no other points of SS lie arbitrarily close to pp. This property implies that in SS cannot approach pp from distinct points, limiting the ways convergence can occur within SS. Isolated points are negligible in integration theory over R\mathbb{R}. Any finite or countable collection of isolated points has zero, and thus functions that differ from a continuous (hence Riemann integrable) function only at such points remain Riemann integrable with the same value. More generally, a on [a,b][a, b] is Riemann integrable its set of discontinuities has measure zero, so discontinuities at isolated points do not obstruct integrability. In Fourier analysis, isolated points in the of certain operators correspond to eigenvalues with associated . For a on a , such as those arising in Fourier multipliers or operators on L2(R)L^2(\mathbb{R}), an isolated point in the belongs to the point spectrum and is thus an eigenvalue admitting a corresponding that spans the eigenspace.

In topology

In , sets consisting entirely of isolated points equip the discrete topology, where every singleton is open, allowing each point to be separated by a neighborhood containing no other points of the set. Such discrete components arise in manifold theory, particularly at isolated singularities, where the local topology near the singularity resembles a over the link of the singular point, influencing the global manifold structure through Milnor fibrations and the classification of exotic spheres. For instance, the link of an isolated complex singularity imposes topological restrictions, such as homotopy equivalence to certain 3-manifolds, aiding in the study of manifold invariants like spin structures. Scattered spaces provide a key application, defined as topological spaces where every non-empty subset contains an isolated point, ensuring a hierarchical into levels of isolated points via . Ordinal spaces, such as the first uncountable ordinal ω1\omega_1, exemplify scattered spaces, as their yields well-ordered bases where subsets inherit isolated points from initial segments. This property is hereditary, meaning every subspace of a scattered remains scattered, preserving the absence of dense-in-itself subsets across subspaces. The isolation property extends to Baire category analysis, where isolated points inform the structure of meager sets; in complete metric spaces lacking isolated points, the guarantees uncountability, highlighting how isolated points disrupt uniformity in category-theoretic classifications. In , isolated points influence s, such as the Brouwer fixed-point theorem on disks, which extends to spheres via degree considerations. Specifically, in , for continuous maps f:Sn→Snf: S^n \to S^n with isolated fixed points, the topological degree equals the sum of local indices at those points, linking local isolation to global invariants like the Lefschetz number.

References

  1. http://pirate.shu.edu/~wachsmut/ira/topo/defs/specldef.html
  2. https://users.math.msu.edu/users/yanb/chapter3.pdf
  3. https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch5.pdf
  4. https://math.wvu.edu/~jwojciec/teaching_files/2024_Spring-581/node-3.html
  5. https://proofwiki.org/wiki/Definition:Topological_Space
  6. https://ncatlab.org/nlab/show/isolated+point
  7. https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Isolated_Point
  8. https://www2.math.upenn.edu/~kazdan/508F14/Notes/basic-defns.pdf
  9. https://people.math.harvard.edu/~ctm/home/text/class/harvard/131/13/html/home/course/course.pdf
  10. https://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf
  11. https://www.math.tugraz.at/~ganster/lv_topologie_ss_2019/exercise_sheet_1.pdf
  12. http://pfister.ee.duke.edu/courses/ece586/notes_ch2.pdf
  13. https://math.gmu.edu/~dwalnut/lec02.pdf
  14. http://mathmatique.com/real-analysis/local-metric-topology/isolated-points-and-limit-points
  15. https://realanalysis.blog/docs/francis-su-analysis-lectures/limit-points/
  16. https://math.ucla.edu/~biskup/131ah.1.23w/PDFs/ch16.pdf
  17. https://www.cis.upenn.edu/~cis6100/cis61011toprev.pdf
  18. https://www.rexresearch1.com/TopologyLibrary/CounterexamplesTopologySteen.pdf
  19. https://mathweb.ucsd.edu/~nlibman/Every_Counterexample_Final.pdf
  20. https://www.bhu.ac.in/research_pub/jsr/Volumes/JSR_65_02_2021/10.pdf
  21. https://www.math.uci.edu/~brusso/220C040908.pdf
  22. https://users.math.msu.edu/users/jeffrey/421/measurezero.pdf
  23. https://link.springer.com/chapter/10.1007/978-1-4612-0741-2_6
  24. https://mathworld.wolfram.com/DiscreteSet.html
  25. https://link.springer.com/book/10.1007/3-7643-7395-4
  26. https://projecteuclid.org/ebooks/advanced-studies-in-pure-mathematics/Complex-Analytic-Singularities/chapter/Topological-Restrictions-on-the-Links-of-Isolated-Complex-Singularities/10.2969/aspm/00810095.pdf
  27. https://web.mat.bham.ac.uk/C.Good/research/pdfs/nests.pdf
  28. https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/
  29. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Add your contribution
Related Hubs
User Avatar
No comments yet.