1 Open Delta neighborhoods

Neighborhood is a set N.

\[N(x_0,\delta) = \{x\in\mathbb{R}^n\ |\ \lvert x-x_0 \rvert < \delta \}\]

• 1-D neigborhood : Interval
• 2-D neighborhood: points in circle
• 3-D neighborhood: points in ball

2 Interior Point - Ball inside Ball

Metric_Space := (M,dist) :: (Set,binary_operation)

Fields of knowledge as Metric Spaces (M,d)

• Metric Space of Proofs in Math
  \((M,d) := (math,dist_{math})\)
• Metric Space of Reactions in Chemistry
  \((M,d) := (chem,dist_{chem})\)

Mapping metric spaces of knowledge AKA analogies

• Analogies as a mapping between Metric Spaces of Knowledge
  • “Reagents is-to Reaction” as “Antecedents is-to succedent”
  • \(dist_{chem}(reagents,reaction) \approx dist_{math}(antecedents,succedent)\)

3 Limit aka Limit point

• Given Metric Space \((M,d)\)
• \(S \subseteq M\)
• p is a limit of S iff \(\exists p_n :: sequence\) s.t. \(p_n converges to p\)

4 Open Set

Let A = (0,0) OpenSet((1,2)) = {(x,y) | (x-1)^2 + (y-2)^2 < r^2 } Boundary((1,2)) = {(x,y) | (x-1)^2 + (y-2)^2 = r^2} ClosedSet((1,2)) = OpenSet((1,2)) + Boundary((1,2))

5 Topological Spaces

Topology(X) = subsets of X s.t.
if a, b are subsets in the topology(X) then a UNION b in topology(X) and a INTERSECT b in topology(X). Nullset and set X are in the topology(X) by default.

6 Manifold

• Manifold is a topological space that locally resembles a euclidean space
• An n-dim manifold has a neighborhood that is homeomorphic to a n-dim euclidean space
  • 1-d manifolds include lines and circles
• Hierarchy of manifolds
  • topological manifold - describe continuous functions
    • differentiable manifold - describe differentiable functions
      • riemannian manifold - inner products defined on tangent space allowing us to define angles,length,depth volume
        • statistical manifold - describe the space of probability distributions