Editing Topology (section)

== <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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