General Topology

Topology is the study of properties that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

Metric Spaces

A metric space $(X, d)$ is a set $X$ with a distance function $d: X \times X \to \mathbb{R}$ satisfying:

Positivity: $d(x, y) \ge 0$ , and $d(x, y) = 0 \iff x = y$ .
Symmetry: $d(x, y) = d(y, x)$ .
Triangle Inequality: $d(x, z) \le d(x, y) + d(y, z)$ .

Interactive Lab

def check_metric_axioms(X, d):
  for x in X:
      for y in X:
          # Positivity
          if d(x, y) < 0: return False, "Negative distance"
          if x != y and d(x, y) == 0: return False, "Zero distance for distinct points"
          # Symmetry
          if d(x, y) != d(y, x): return False, "Symmetry failed"
          for z in X:
              # Triangle Inequality
              if d(x, z) > d(x, y) + d(y, z) + 1e-9:
                  return False, f"Triangle inequality failed for {x}, {y}, {z}"
  return True, "Valid Metric"

# Manhattan Distance
def taxicab(p1, p2):
  return abs(p1[0] - p2[0]) + abs(p1[1] - p2[1])

points = [(0,0), (1,1), (2,3), (10, 0)]
print(f"Taxicab metric: {check_metric_axioms(points, taxicab)}")

# Discrete Metric
def discrete(p1, p2):
  return 0 if p1 == p2 else 1

print(f"Discrete metric: {check_metric_axioms(points, discrete)}")

python

1def check_metric_axioms(X, d):

2 for x in X:

3 for y in X:

4 # Positivity

5 if d(x, y) < 0: return False, "Negative distance"

6 if x != y and d(x, y) == 0: return False, "Zero distance for distinct points"

7 # Symmetry

8 if d(x, y) != d(y, x): return False, "Symmetry failed"

9 for z in X:

10 # Triangle Inequality

11 if d(x, z) > d(x, y) + d(y, z) + 1e-9:

12 return False, f"Triangle inequality failed for {x}, {y}, {z}"

13 return True, "Valid Metric"

15# Manhattan Distance

16def taxicab(p1, p2):

17 return abs(p1[0] - p2[0]) + abs(p1[1] - p2[1])

19points = [(0,0), (1,1), (2,3), (10, 0)]

20print(f"Taxicab metric: {check_metric_axioms(points, taxicab)}")

22# Discrete Metric

23def discrete(p1, p2):

24 return 0 if p1 == p2 else 1

26print(f"Discrete metric: {check_metric_axioms(points, discrete)}")

Topological Spaces

A topological space $(X, \tau)$ is a set $X$ with a collection of subsets $\tau$ (the open sets) such that:

$\emptyset \in \tau$ and $X \in \tau$ .
Arbitrary unions of sets in $\tau$ are in $\tau$ .
Finite intersections of sets in $\tau$ are in $\tau$ .

Interactive Lab

def is_topology(X, tau):
  X_set = set(X)
  tau_sets = [set(s) for s in tau]
  
  # Empty set and X
  if set() not in tau_sets or X_set not in tau_sets:
      return False, "Missing empty set or X"
  
  # Finite Intersections
  import itertools
  for s1, s2 in itertools.combinations(tau_sets, 2):
      if s1.intersection(s2) not in tau_sets:
          return False, f"Intersection of {s1} and {s2} not in tau"
          
  # Binary Unions (covers arbitrary unions for finite sets)
  for s1, s2 in itertools.combinations(tau_sets, 2):
      if s1.union(s2) not in tau_sets:
          return False, f"Union of {s1} and {s2} not in tau"
          
  return True, "Valid Topology"

X = [1, 2, 3]
indiscrete = [[], [1, 2, 3]]
print(f"Indiscrete Topology: {is_topology(X, indiscrete)}")

overlap = [[], [1], [2], [1, 2, 3]]
print(f"Partial Overlap: {is_topology(X, overlap)}")

python

1def is_topology(X, tau):

2 X_set = set(X)

3 tau_sets = [set(s) for s in tau]

5 # Empty set and X

6 if set() not in tau_sets or X_set not in tau_sets:

7 return False, "Missing empty set or X"

9 # Finite Intersections

10 import itertools

11 for s1, s2 in itertools.combinations(tau_sets, 2):

12 if s1.intersection(s2) not in tau_sets:

13 return False, f"Intersection of {s1} and {s2} not in tau"

15 # Binary Unions (covers arbitrary unions for finite sets)

16 for s1, s2 in itertools.combinations(tau_sets, 2):

17 if s1.union(s2) not in tau_sets:

18 return False, f"Union of {s1} and {s2} not in tau"

20 return True, "Valid Topology"

22X = [1, 2, 3]

23indiscrete = [[], [1, 2, 3]]

24print(f"Indiscrete Topology: {is_topology(X, indiscrete)}")

26overlap = [[], [1], [2], [1, 2, 3]]

27print(f"Partial Overlap: {is_topology(X, overlap)}")

Continuity

A function $f: X \to Y$ is continuous if the preimage of every open set in $Y$ is open in $X$ .

Formally: $f$ is continuous if $\forall V \in \tau_Y, f^{-1}(V) \in \tau_X$ .

In a Metric Space, this is equivalent to the $\epsilon-\delta$ definition: $f$ is continuous at $x_0$ if: $\forall \epsilon > 0, \exists \delta > 0 \text{ such that } d_X(x, x_0) < \delta \implies d_Y(f(x), f(x_0)) < \epsilon$

In a discrete metric space, which sets are open?

Compactness and Connectedness

A space is compact if every open cover has a finite subcover. A space is connected if it cannot be partitioned into two disjoint non-empty open sets.

General Topology

General Topology

Metric Spaces

Topological Spaces

Continuity

In a discrete metric space, which sets are open?

Compactness and Connectedness

The Heine-Borel theorem states that subsets of R^n are compact if and only if they are: