Editing Topology

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{{BloomIntro}}
Topology is the branch of mathematics that studies properties of spaces that are preserved under continuous deformations — stretching, bending, and twisting, but not tearing or gluing. A topologist famously cannot distinguish a coffee mug from a donut (both have one hole), but can distinguish either from a sphere (which has none). Topology captures the most fundamental, qualitative features of geometric objects: connectedness, compactness, continuity, and the number and type of "holes." It has transformed mathematics by providing the language of continuity used across analysis, geometry, algebra, and physics. Algebraic topology — which uses algebraic invariants to classify topological spaces — is one of the deepest and most beautiful areas of 20th century mathematics.
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== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Topology''' — The mathematical study of properties of spaces preserved under continuous deformations (homeomorphisms).
* '''Topological space''' — A set X with a collection of subsets (the open sets) satisfying: ∅ and X are open; arbitrary unions of open sets are open; finite intersections are open.
* '''Homeomorphism''' — A bijective continuous map with a continuous inverse; the isomorphism of topology; topologically equivalent spaces.
* '''Open set / closed set''' — Fundamental building blocks; open sets are the "neighborhoods" defining the topology.
* '''Basis (topology)''' — A collection of open sets from which all open sets can be generated by unions.
* '''Continuity (topological)''' — f is continuous iff preimages of open sets are open; generalizes ε-δ continuity.
* '''Connectedness''' — A space is connected if it cannot be partitioned into two disjoint non-empty open sets.
* '''Path-connectedness''' — Any two points can be joined by a continuous path; stronger than connectedness.
* '''Compactness''' — Every open cover has a finite subcover; generalizes closed-and-bounded in ℝⁿ.
* '''Homotopy''' — A continuous deformation of one map (or space) into another.
* '''Fundamental group π₁''' — The group of loops (up to homotopy) based at a point; captures 1-dimensional holes.
* '''Homology groups''' — Algebraic invariants measuring n-dimensional holes in a space; H₀, H₁, H₂, ...
* '''Euler characteristic (χ)''' — A topological invariant: χ = V − E + F for polyhedra; χ(sphere) = 2, χ(torus) = 0.
* '''Manifold''' — A topological space that locally looks like Euclidean space; curves are 1-manifolds, surfaces are 2-manifolds.
* '''Genus''' — The number of handles on a surface; torus has genus 1, sphere has genus 0.
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== <span style="color: #FFFFFF;">Understanding</span> ==
Topology's central insight: many geometric properties are "accidents" of a particular embedding, while topological properties are intrinsic. Two coffee mugs are geometrically different but topologically the same.

'''Point-set topology''': The foundational layer. The open set axioms generalize familiar properties of ℝ to arbitrary spaces. Key theorems: the intermediate value theorem (a continuous function on a connected interval that takes two values takes all values between them), the extreme value theorem (a continuous function on a compact space attains its maximum and minimum), Tychonoff's theorem (any product of compact spaces is compact — requires Axiom of Choice). Metric spaces, topological groups, and function spaces all live in this framework.

'''Algebraic topology''': Instead of asking "what does this space look like?" we ask "what algebraic invariants distinguish spaces?" The fundamental group π₁(X) measures 1-dimensional loops: π₁(circle) = ℤ, π₁(sphere) = 0, π₁(torus) = ℤ × ℤ. Higher homotopy groups and homology groups measure higher-dimensional holes. The Euler characteristic χ = Σ(-1)^k rank(H_k) is a topological invariant. The classification of surfaces is complete: every compact orientable surface is determined by its genus (number of handles).

'''The Poincaré Conjecture''': Henri Poincaré conjectured (1904) that any simply connected compact 3-manifold without boundary is homeomorphic to the 3-sphere. It resisted all attacks until Grigori Perelman proved it in 2003 using Richard Hamilton's Ricci flow. Perelman declined the Fields Medal and $1M Millennium Prize. The higher-dimensional versions (n ≥ 5) had been proved earlier by Smale (1960, Fields Medal) and Freedman (1982, Fields Medal).

'''Topology in modern mathematics and physics''': Differential topology studies smooth manifolds; algebraic geometry uses sheaf theory and étale cohomology. In physics, topological quantum field theories, topological insulators, and the quantum Hall effect are described by topological invariants. The Atiyah-Singer Index Theorem connects differential operators to topological invariants — one of the deepest results of 20th century mathematics.
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== <span style="color: #FFFFFF;">Applying</span> ==
'''Computational topology — persistent homology:'''
<syntaxhighlight lang="python">
import numpy as np
from itertools import combinations

# Persistent homology computes topological features (connected components,
# loops, voids) at multiple scales — used in topological data analysis (TDA)

def build_vietoris_rips_complex(points: np.ndarray, epsilon: float) -> dict:
    """
    Build Vietoris-Rips complex at scale ε:
    Add simplex {v0, v1, ..., vk} if all pairwise distances ≤ ε.
    """
    n = len(points)
    complex_dict = {'0-simplices': list(range(n)), '1-simplices': [], '2-simplices': []}
    
    # 1-simplices: edges
    for i, j in combinations(range(n), 2):
        if np.linalg.norm(points[i] - points[j]) <= epsilon:
            complex_dict['1-simplices'].append((i, j))
    
    # 2-simplices: triangles
    for i, j, k in combinations(range(n), 3):
        if (np.linalg.norm(points[i] - points[j]) <= epsilon and
            np.linalg.norm(points[j] - points[k]) <= epsilon and
            np.linalg.norm(points[i] - points[k]) <= epsilon):
            complex_dict['2-simplices'].append((i, j, k))
    
    return complex_dict

def euler_characteristic(complex_dict: dict) -> int:
    """χ = V - E + F (Euler-Poincaré formula)."""
    V = len(complex_dict['0-simplices'])
    E = len(complex_dict['1-simplices'])
    F = len(complex_dict['2-simplices'])
    return V - E + F

# Create point cloud: circle in 2D
angles = np.linspace(0, 2*np.pi, 12, endpoint=False)
circle_points = np.column_stack([np.cos(angles), np.sin(angles)])
circle_points += np.random.normal(0, 0.1, circle_points.shape)  # Add noise

# Build complexes at multiple scales
for eps in [0.3, 0.6, 1.0, 2.0]:
    cpx = build_vietoris_rips_complex(circle_points, eps)
    chi = euler_characteristic(cpx)
    n_edges = len(cpx['1-simplices'])
    print(f"ε={eps}: V={len(cpx['0-simplices'])}, E={n_edges}, "
          f"F={len(cpx['2-simplices'])}, χ={chi}")
# At the right scale, χ=0 → we detect the circle (1 connected component, 1 loop)
# χ(circle) = 0; χ(disk) = 1; χ(sphere) = 2

# In practice, use Gudhi or ripser for efficient persistent homology
# import gudhi; rips = gudhi.RipsComplex(points=circle_points, max_edge_length=1.0)
</syntaxhighlight>

; Key results and areas
: '''Point-set topology''' → Hausdorff separation, Tychonoff theorem, Urysohn metrization
: '''Algebraic topology''' → Euler characteristic, fundamental group, homology/cohomology, homotopy groups
: '''Classification theorems''' → Surfaces by genus; 3-manifolds (geometrization, Thurston/Perelman)
: '''Differential topology''' → Morse theory, cobordism, Whitney embedding theorem
: '''Applied topology (TDA)''' → Persistent homology, mapper algorithm, topological data analysis
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== <span style="color: #FFFFFF;">Analyzing</span> ==
{| class="wikitable"
|+ Topological Invariants of Surfaces
! Surface !! χ (Euler char) !! π₁ (fundamental group) !! Genus !! Orientable?
|-
| Sphere S² || 2 || Trivial {e} || 0 || Yes
|-
| Torus T² || 0 || ℤ × ℤ || 1 || Yes
|-
| Double torus || −2 || Complex (genus 2 surface group) || 2 || Yes
|-
| Real projective plane ℝP² || 1 || ℤ/2ℤ || — || No
|-
| Klein bottle || 0 || Non-abelian extension || — || No
|}

'''Famous theorems''': Brouwer Fixed Point Theorem: any continuous map from a closed disk to itself has a fixed point. Hairy Ball Theorem: there is no non-vanishing continuous tangent vector field on a sphere (you can't comb a hairy ball flat). Jordan Curve Theorem: any simple closed curve in the plane divides it into two regions. Borsuk-Ulam Theorem: any continuous map from Sⁿ to ℝⁿ maps some pair of antipodal points to the same point.
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== <span style="color: #FFFFFF;">Evaluating</span> ==
Topology concepts and proofs are assessed through:
# '''Invariance''': is the claimed invariant truly topological (preserved under homeomorphisms)?
# '''Distinguishing power''': does the invariant separate the spaces you want to distinguish?
# '''Computability''': can the invariant be computed from the definition?
# '''Functoriality''': is the construction natural — does it commute with maps in the right way?
# '''Applications''': does the topological framework reveal structure invisible to other approaches?
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== <span style="color: #FFFFFF;">Creating</span> ==
Advanced and applied topology:
# '''Persistent homology for data analysis''': compute topological features of high-dimensional point clouds; detect clusters (H₀), loops (H₁), and voids (H₂) across scales.
# '''Topological data analysis''': apply TDA to brain connectivity, materials structure, genomics data.
# '''Topological quantum computation''': Majorana fermions and anyons — topological invariants protect quantum information from decoherence.
# '''Symplectic topology''': study of phase spaces in classical and quantum mechanics.
# '''K-theory''': generalized cohomology theory with applications to operator algebras, string theory, and the classification of topological phases of matter.

[[Category:Mathematics]]
[[Category:Topology]]
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