Topology - Spaces, Continuity, and Manifolds

Table of Contents

1. Topological Spaces
2. Continuity
3. Compactness
4. Connectedness
5. Metric Spaces
6. Manifolds
7. Homotopy and Fundamental Group
8. Differential Geometry

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Topological Spaces

Definition

Topology $\tau$ on set $X$: Collection of open sets satisfying:

1. $\emptyset, X \in \tau$
2. Union of any collection in $\tau$ is in $\tau$
3. Intersection of finitely many sets in $\tau$ is in $\tau$

$(X, \tau)$ is topological space

Basis

Basis for topology: Collection B where every open set is union of basis elements

Metric topology: Open balls form basis

Product topology: Basis is Cartesian product of basis elements

Closed Sets

Closed set: Complement of open set

Properties:

• Intersection of closed sets closed
• Finite union of closed sets closed

Interior, Closure, Boundary

Interior Int(A): Largest open set contained in A

Closure $\bar{A}$: Smallest closed set containing A

Boundary $\partial A$: $\bar{A} - \text{Int}(A) = \text{Closure} \cap \text{Closure of complement}$

Subspace Topology

$Y \subseteq X$: Open in $Y = U \cap Y$ for $U$ open in $X$

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Continuity

Definition

$f: X \to Y$ continuous if $f^{-1}(U)$ open in $X$ for all open $U \subseteq Y$

Equivalent characterizations:

• $f^{-1}(F)$ closed for closed $F$
• For neighborhoods $V$ of $f(x)$, $f^{-1}(V)$ is neighborhood of $x$

Continuity at a Point

$f$ continuous at $x$ if for neighborhood $V$ of $f(x)$, there exists neighborhood $U$ of $x$ with $f(U) \subseteq V$

Homeomorphism

Bijective continuous function with continuous inverse

$f$ is homeomorphism $\iff$ $f, f^{-1}$ continuous

Topologically equivalent spaces

Properties Preserved by Homeomorphism

• Compactness
• Connectedness
• Separation properties (but not metric properties)

Continua

Continuous image of compact connected space

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Compactness

Open Cover

Open cover of A: Collection of open sets whose union contains A

Compact

A compact if every open cover has finite subcover

Heine-Borel Theorem: In $\mathbb{R}^n$ with standard topology, compact $\iff$ closed and bounded

Properties of Compact Sets

• Closed subsets of compact spaces are compact
• Continuous image of compact set is compact
• Product of compact sets is compact (Tychonoff)

Sequential Compactness (Metric Spaces)

Sequentially compact: Every sequence has convergent subsequence

For metric spaces: Compact $\iff$ sequentially compact

Limit Point Compact

Every infinite subset has limit point

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Connectedness

Connected

Space connected if cannot be written as disjoint union of nonempty open sets

Alternative: No nontrivial open and closed subset

Path Connected

Space path-connected if any two points connected by continuous path

Path-connected $\implies$ connected (converse not always true)

Components

Connected component: Maximal connected subset

Path component: Maximal path-connected subset

Arc Connected

Path with injective function

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Metric Spaces

Definition

Metric $d$ on $X$: Function $d: X \times X \to \mathbb{R}$ satisfying:

1. $d(x,y) \geq 0, = 0 \iff x = y$
2. $d(x,y) = d(y,x)$ (symmetric)
3. $d(x,z) \leq d(x,y) + d(y,z)$ (triangle inequality)

$(X, d)$ is metric space

Examples

Euclidean: $d(x,y) = ||x - y||$ Sup norm: $d(f,g) = \sup|f(x) - g(x)|$ Discrete metric: $d(x,y) = 0$ if $x=y$, $=1$ otherwise

Balls and Neighborhoods

Open ball: $B_r(x) = \{y : d(x,y) < r\}$

Closed ball: $\bar{B}_r(x) = \{y : d(x,y) \leq r\}$

Complete Metric Spaces

Complete: Every Cauchy sequence converges

Banach space: Complete normed space

Examples: $\mathbb{R}^n$, $C[a,b]$ with sup norm

Completion

Every metric space can be completed

Contraction Mapping

$T: X \to X$ is contraction if $d(Tx, Ty) \leq \alpha d(x,y)$ for some $\alpha < 1$

Fixed point theorem: Contraction on complete metric space has unique fixed point

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Manifolds

Topological Manifold

$n$-manifold: Hausdorff space with each point having neighborhood homeomorphic to $\mathbb{R}^n$

Chart: $(U, \phi)$ where $U \subseteq M$ open, $\phi: U \to \mathbb{R}^n$ homeomorphism

Atlas: Collection of charts covering $M$

Smooth Manifold

$C^\infty$ manifold: Charts chosen so transition functions $\phi_i \circ \phi_j^{-1}$ are $C^\infty$

Differentiable manifold

Orientability

Manifold is orientable if can consistently choose orientation at each point

Non-orientable: Möbius band

Tangent Space

At point $p$: $T_pM = \{\text{directional derivatives at } p\}$

Dimension: Same as manifold

Vector Fields

Smooth assignment of tangent vector to each point

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Homotopy and Fundamental Group

Homotopy

$f_0$ and $f_1$ homotopic if continuous deformation $f_t$ between them

Map: $H: X \times [0,1] \to Y, H(x,0) = f_0(x), H(x,1) = f_1(x)$

Homotopy Equivalence

Spaces homotopy equivalent if can deform one to other

Weaker than homeomorphism: Invariant under continuous deformation

Fundamental Group $\pi_1(X, x_0)$

Group of loops based at $x_0$ up to homotopy

Composition: Concatenation of loops

Identity: Constant loop

Inverse: Reverse loop

Abelian for many spaces

Covering Spaces

$p: E \to B$ covering map if locally homeomorphism

Universal cover: Simply connected covering space

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Differential Geometry

Smooth Manifold

$n$-dimensional manifold locally like $\mathbb{R}^n$

Tangent bundle $TM = \cup_{p \in M} T_pM$

Differential Forms

$k$-form on M: Antisymmetric multilinear map on $k$ tangent vectors

Exterior derivative $d$: Generalizes gradient, curl, divergence

Integration on Manifolds

Integrate $k$-forms over $k$-dimensional submanifolds

Stokes’ Theorem:

$\int_M d\omega = \int_{\partial M} \omega$

Generalizes:

• Fundamental theorem of calculus
• Green’s theorem
• Divergence theorem

Curvature

Gaussian curvature $K$: Intrinsic property (Gauss’s Theorema Egregium)

Mean curvature $H$: Extrinsic property

Riemannian metric $g$: Inner product on tangent spaces

Levi-Civita connection $\nabla$: Unique torsion-free metric connection

Geodesics

Length-minimizing curves with respect to metric

Euler-Lagrange equation for energy functional

Geodesic equation: $\nabla_{\gamma'}\gamma' = 0$

Metric Tensor

$g_{ij}(x)$ describes local geometry

Riemann tensor: Measures intrinsic curvature

Einstein equations: Relate geometry to matter

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Next: Abstract Algebra or Fourier Analysis

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Last updated: Comprehensive topology reference covering spaces, manifolds, and differential geometry.