Table of Contents

Introduction and Definitions

  1. Topology:
    is a mathematical field concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing
  2. Topological Space:
    is defined as a set of points \(\mathbf{X}\), along with a set of neighbourhoods (sub-sets) \(\mathbf{T}\) for each point, satisfying the following set of axioms relating points and neighbourhoods:
    • \(\mathbf{T}\) is the Open Sets:
      1. The Empty Set \(\emptyset\) is in \(\mathbf{T}\)
      2. \(\mathbf{X}\) is in \(\mathbf{T}\)
      3. The Intersection of a finite number of Sets in \(\mathbf{T}\) is, also, in \(\mathbf{T}\)
      4. The Union of an arbitrary number of Sets in \(\mathbf{T}\) is, also, in \(\mathbf{T}\)
    • \(\mathbf{T}\) is the Closed Sets:
      1. The Empty Set \(\emptyset\) is in \(\mathbf{T}\)
      2. \(\mathbf{X}\) is in \(\mathbf{T}\)
      3. The Intersection of an arbitrary number of Sets in \(\mathbf{T}\) is, also, in \(\mathbf{T}\)
      4. The Union of a finite number of Sets in \(\mathbf{T}\) is, also, in \(\mathbf{T}\)
  3. Homeomorphism:
    Intuitively, a Homeomorphism or Topological Isomorphism or bi-continuous Function is a continuous function between topological spaces that has a continuous inverse function.
    Mathematically, a function \({\displaystyle f:X\to Y}\) between two topological spaces \({\displaystyle (X,{\mathcal {T}}_{X})}\) and \({\displaystyle (Y,{\mathcal {T}}_{Y})}\) is called a Homeomorphism if it has the following properties:
    • \(f\) is a bijection (one-to-one and onto)
    • \(f\) is continuous
    • the inverse function \({\displaystyle f^{-1}}\) is continuous (\({\displaystyle f}\) is an open mapping).

    i.e. There exists a continuous map with a continuous inverse

  4. Maps and Spaces:
    Map Space Preserved Property
    Linear Map Vector Space Linear Structure: \(f(aw+v) = af(w)+f(v)\)
    Group Homomorphism Group Group Structure: \(f(x \ast y) = f(x) \ast f(y)\)
    Continuous Map Topological Space Openness/Closeness: \(f^{-1}(\{\text{open}\}) \text{ is open}\)
    Smooth Map Topological Space  
  5. Smooth Maps:
    • Continuous:
    • Unique Limits:
  6. Hausdorff:

Point-Set Topology

  1. Open Set:
    A set \(\chi \subseteq \mathbf{R}^n\) is said to be open if for any point \(x \in \chi\) there exist a ball centered in \(x\) which is contained in \(\chi\).
    Precisely, for any \(x \in \mathbf{R}^n\) and \(\epsilon > 0\) define the Euclidean ball of radius \(r\) centered at \(x\):
    \[B_\epsilon(x) = {z : \|z − x\|_2 < \epsilon}\]
    Then, \(\chi \subseteq \mathbf{R}^n\) is open if
    \[\forall x \: \epsilon \: \chi, \:\: \exists \epsilon > 0 : B_\epsilon(x) \subset \chi .\]
    Equivalently,
    • A set \(\chi \subseteq \mathbf{R}^n\) is open if and only if \(\chi = int\; \chi\).
    • An open set does not contain any of its boundary points.
    • A closed set contains all of its boundary points.
    • Unions and intersections of open (resp. closed) sets are open (resp. closed).
  2. Closed Set:
    A set \(\chi \subseteq \mathbf{R}^n\) is said to be closed if its complement \(\mathbf{R}^n \text{ \ } \chi\) is open.
  3. Interior of a Set:
    The interior of a set \(\chi \subseteq \mathbf{R}^n\) is defined as
    \[int\: \chi = \{z \in \chi : B_\epsilon(z) \subseteq \chi, \:\: \text{for some } \epsilon > 0 \}\]
  4. Closure of a Set:
    The closure of a set \(\chi \subseteq \mathbf{R}^n\) is defined as
    \[\bar{\chi} = \{z ∈ \mathbf{R}^n : \: z = \lim_{k\to\infty} x_k, \: x_k \in \chi , \: \forall k\},\]

    i.e., the closure of \(\chi\) is the set of limits of sequences in \(\chi\).

  5. Boundary of a Set:
    The boundary of X is defined as
    \[\partial \chi = \bar{\chi} \text{ \ } int\: \chi\]
  6. Bounded Set:
    A set \(\chi \subseteq \mathbf{R}^n\) is said to be bounded if it is contained in a ball of finite radius, that is if there exist \(x \in \mathbf{R}^n\) and \(r > 0\) such that \(\chi \subseteq B_r(x)\).
  7. Compact Set:
    A set \(\chi \subseteq \mathbf{R}^n\) is compact \(\iff\) it is Closed and Bounded.
  8. Relative Interior [\(\operatorname{relint}\)]:
    We define the relative interior of the set \(\chi\), denoted \(\operatorname{relint} \chi\), as its interior relative to \(\operatorname{aff} C\):
    \[\operatorname{relint} \chi = \{x \in \chi : \: B(x, r) \cap \operatorname{aff} \chi \subseteq \chi \text{ for some } r > 0\},\]
    where \(B(x, r) = \{y : ky − xk \leq r\}\), the ball of radius \(r\) and center \(x\) in the norm \(\| · \|\).
  9. Relative Boundary:
    We can then define the relative boundary of a set \(\chi\) as \(\mathbf{cl} \chi \text{ \ } \operatorname{relint} \chi,\) where \(\mathbf{cl} \chi\) is the closure of \(\chi\).

Manifolds

  1. Manifold:
    is a topological space that locally resembles Euclidean space near each point

    i.e. around every point, there is a neighborhood that is topologically the same as the open unit ball in \(\mathbb{R}^n\)

  2. Smooth Manifold:
    A topological space \(M\) is called a \(n\)-dimensional smooth manifold if:
    • Is is Hausdorff
    • It is Second-Countable
    • It comes with a family \(\{(U_\alpha, \phi_\alpha)\}\) with:
      • Open sets \(U_\alpha \subset_\text{open} M\)
      • Homeomorphisms \(\phi_\alpha : U_\alpha \rightarrow \mathbb{R}^n\)
        such that \({\displaystyle M = \bigcup_\alpha U_\alpha}\)
        and given \({\displaystyle U_\alpha \cap U_\beta \neq \emptyset}\) the map \(\phi_\beta \circ \phi_\alpha^{-1}\) is smooth