4 Topology
definition (Topology) A topology, \(\mathcal{T}\), is a collection of subsets of X that satisfies the following conditions:
- Existence of the set. That is, \(X\) is in \(\mathcal{T}\).
- Existence of the empty set. That is, \(\emptyset\) is in \(\mathcal{T}\).
- Closure under finite intersections. That is, \(U_1 \cap \dots \cap U_n\) is an element of \(\mathcal{T}\).
- Closure under arbitrary unions. That is, \(\bigcup_{\alpha \in A} U_\alpha\) is also an element of \(\mathcal{T}\).
where
- \(U_i\) are elements of \(\mathcal{T}\)
- \(X\) is a set
4.0.1 Manifolds
There are two main types of manifolds, generally: topological and smooth manifolds. We will start with an informal definition of a topological manifold.
If a surface is a generalization of a curve, then a topological manifold is the generalization of both surfaces and curves.
Here’s another way to understand a topological manifold.
A curve is a one-dimension manifold.
A surface is a two-dimension manifold.
Beyond two dimensions, we just call these things n-manifold, where \(n\) is the number of dimensions.
Let’s look at a more formal definition of a manifold.
definition (Topological Manifold) A topological manifold is a topological space, \(M\), that satisfies the following conditions:
- Locally Euclidean of Dimension \(n\)
- Hausdorff Space
- Second Countable
definition (Smooth Manifold)
4.0.2 Topological Spaces
definition (Topological Space) A topological space is a pair \((X, \mathcal{T})\) on \(X\), where
- \(X\) is a set
- \(\mathcal{T}\) is a topology
definition (Hausdorff Space) A Hausdorff space is a topological space, \(X\), that satisfies the following condition:
- Disjoint neighborhoods: \(U_1 \cap U_2 = \empty\), for any pair of points \(p_1,p_2 \in X\)
where
- \(U_1, U_2\) are neighborhoods
- \(p_1,p_2 \in X\) are distinct points
- \(X\) is a topological space
definition (Neighborhood) A property of a subset \(N\) of a topological space \(X\) with respect to a point \(p \in X\), where the following condition applies:
- There exists an open set \(U\) such that \(p \in U \subseteq N\).
where
- \(X\) is a topological space.
- \(p\) is a point in \(X\).
- \(N\) is a subset of \(X\).
- \(U\) is an open subset of \(X\).
4.0.3 Homeomorphism
A description of two spaces that have the same topological structure.
Now about something called topological invariants.
Properties (such as numbers, groups, matrices, or vector spaces) of a topological space that are preserved by homeomorphisms. If two spaces possess different invariants, they cannot be topologically equivalent. Examples include the fundamental group, homology groups, and the Euler characteristic.
definition (Homeomorphism) A function between topological spaces \(f: \mathbf{U} \to \mathbf{V}\) is called a homeomorphism if the following conditions hold.
- \(f\) is a bijection and continuous
- \(f^{-1}\) is continuous
where
- \(f^{-1}\) is the inverse of \(f\)
- \(\mathbf{U}\), \(\mathbf{V}\) are topological spaces.
4.0.4 Open Sets
The open set is central to topology, and it is required for definitions. We begin with a definition and then an example.
definition (Open Set) A subset \(U\) of a topological space, \((X,\mathcal{T})\), where the following condition applies:
- \(U \in \mathcal{T}\)
where
- \(X\) is a set
- \(\mathcal{T}\) is a topology on \(X\)
- \(U \subseteq X\)
example (Open Set) Let
\[ X = \{1,2,3\} \]
\[ \mathcal{T} = \{ \emptyset, \{1\}, \{1,2\}, X \}. \]
Then, \((X,\mathcal{T})\) is a topological space, where
\[ (X,\mathcal{T}) = \left( \{1,2,3\}, \{ \emptyset, \{1\}, \{1,2\}, \{1,2,3\} \} \right) \].
Let
\[ U = \{1,2\}. \]
Since
\[ U \in \mathcal{T}, \]
it follows that \(U\) is an open set.
Therefore,
\[ \{1,2\} \]
is an open set in \((X,\mathcal{T})\).
4.1 APPENDIX
4.1.1 Preimages and Inverse Functions
definition (Inverse function) A function from the codomain of a function to its domain,
\[ f^{-1} : Y \rightarrow X, \]
where the following condition applies:
- \(f^{-1}(y) = x \iff f(x) = y\).
where
- \(f : X \rightarrow Y\) is a one-to-one and onto function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(x \in X, y \in Y\).
definition (Preimage) A subset of the domain of a function,
\[ f^{-1}(Y_{subset}) \subseteq X, \]
where the following condition applies:
- \(f^{-1}(Y_{subset}) = \{x \in X : f(x) \in Y_{subset}\}\).
where
- \(f : X \rightarrow Y\) is a function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(Y_{subset} \subseteq Y\).
definition (Continuity) A property of a map between open subsets, \(f: X \rightarrow Y\), where the following conditions apply:
- The preimage of every open subset of \(Y\) is an open subset of \(X\).
where
- \(X, Y \subseteq \mathbb{R}^n\)
- \(f^{-1}(U) = \{x \in X : f(x) \in U\}\)
- \(f^{-1} : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)\) is the preimage map from subsets of \(Y\) to subsets of \(X\).
- \(U \subseteq Y\)
- \(\mathcal{P}(X)\) is the power set of \(X\)
definition (Surjective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For every element \(y\) in \(Y\), there exists an element \(x\) in \(X\) such that \(F(x) = y\).
where
- \(X\) is the domain of \(F\)
- \(Y\) is the co-domain of \(F\)
- \(F(X) = Y\) (the range of the function is equal to its co-domain).
- \(F(X) = \{y \in Y : y = F(x) \text{ for some } x \in X\}\) is the image of \(X\) under \(F\).
definition (Injective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For all elements \(x_1\) and \(x_2\) in \(X\), if \(F(x_1) = F(x_2)\), then \(x_1 = x_2\).
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\).
- The condition is logically equivalent to saying that if \(x_1 \neq x_2\), then \(F(x_1) \neq F(x_2)\).
- For an injective function, each element of the co-domain is the image of at most one element of the domain.
definition (Bijective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following conditions apply:
- The function \(F\) is injective
- The function \(F\) is surjective
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\)
4.2 GLOSSARY
Smooth Manifold: -
Locally: -
Homeomorphic: A description of two spaces that have the same topological structure.
Genus
Holes: An informal synonym for genus.
Tori A surface that looks like a donut. It has one hole.
Topological Structure: -
Continuous Function: -
Differential Form A function assigning an alternating k-linear function to each point of an open set or manifold.
Surface A 2-dimensional manifold, an object modeled locally on \(\mathbb{R}^2\), specifying points with two independent parameters.
Piecewise Smooth A property of curves or surfaces composed of finitely many smooth segments meeting at corners or edges.
Oriented Surface A surface with a consistent choice of orientation at each point, typically specified by a nowhere-vanishing 2-form.
Smooth Surface A 2-dimensional manifold equipped with a differentiable structure, allowing for calculus and well-defined tangent spaces.
Surface Normal:
Piecewise Smooth Simple Closed Curve A non-self-intersecting closed path consisting of finitely many differentiable segments joined end-to-end.
Open Set: A set that is open in a topological space.
Open Set (Expanded): A subset of a topological space that contains a neighborhood around every one of its points.
Differentiable Function -
Smooth Function -
K-Covector An alternating \(k\)-linear function on a vector space, also known as a multicovector of degree \(k\).
K-Covector field = K-Form A function assigning a \(k\)-covector to each point of a manifold, also called a differential \(k\)-form.
K-Tensor A multilinear function from the \(k\)-fold product of a vector space to the real numbers.
K-Fold A term denoting the repetition of a set or space \(k\) times, typically within a Cartesian product.
K-linear A synonym for multilinear function.
K-Fold product The Cartesian product of \(k\) copies of a set \(X\), often denoted simply as \(X^k\).
Bijection A function that is both injective and surjective, also referred to as a one-to-one correspondence.
Hausdorff Space A topological space where any two distinct points can be separated by disjoint open neighborhoods.
Onto A synonym for surjective function.
One-to-One A synonym for an injective function.
Covariant vector = Covector = 1-form