Notes on Topology

Topological Spaces and Topologies

Given a set X and a collection of its subsets τ (this τ is a topology on X), a topological space on X is denoted by (X,τ) and follows the rules below.

• X and the empty set are contained in τ.

• Any union of subsets in τ is contained in τ. This can be an infinite number of unions.

• Any finite intersection of subsets in τ is contained in τ.

Neighborhoods 

• Given a topological space (X,τ) containing an element A, the neighborhood N_A is any subset of X including A such that N_A contains an open set of τ.
• Often, neighborhoods are defined around a point p in X. In these cases, the neighborhood N_p is any subset of X including p such that N_p contains an open set of τ.
• To visualize neighborhoods, consider a subset V of a plane X. The subset V contains a point p. If an arbitrarily small disk around p fits inside V, then V is a neighborhood of p.
• Note that, for a given set S and any point p on the boundary of the set, that set cannot be a neighborhood of p.

Open Sets and Closed Sets 

• In practice, open sets do not include their boundaries and closed sets do include their boundaries. However, more complicated definitions exist.
• The union of any number of open sets is itself open.
• The intersection of a finite number of open sets is itself open.
• The complement of an open set relative to the rest of the space is a closed set.
• Some sets may be both open and closed. Examples of such sets are the given entire topological space X and the empty set.

Closure, Interior, and Boundary of a Set

• Informally, the boundary of a subset S includes the points on its “outline.”
• Slightly more formally, the boundary includes the points in S for which no open ball will be entirely inside S.

• The interior of a subset S includes all points in S that are not part of its boundary.
• The closure of a subset S on a topology is defined as the union of the boundary points of S and the interior points of S.

Topological Bases

• A basis B for a topology X is a family of subsets of X such that every open subset of X is the union of some members of B.
• The empty set is the union of any nonoverlapping elements of B.
• Any intersection of base elements is another base element.
• Many bases may generate the same topology.
• A common example of a basis is the set of all possible open balls (in 2D they actually are disks, though they are called balls no matter the dimensionality) that union to form a plane.
• Below, just some of the (infinite) open balls forming the basis for a topology consisting of a 2D shape are illustrated.

Continuous Functions between Topological Spaces: Homeomorphisms

• Functions between topological spaces generalize the notion of real and complex-valued mappings to any rule that assigns abstract objects in a domain to abstract objects in a codomain.
• Continuous functions between topological spaces are called homeomorphisms when the inverse of the given mapping is also continuous.
• Given a mapping from a topological space (X,τ) to a topological space (Y,τ), the mapping is continuous if and only if all open sets in the domain map to open sets in the codomain. That is, the preimage of any open set in the codomain must be open in the domain.
• Another property to note is that homeomorphisms are bijective (each element in the domain maps to exactly one element in the codomain).

• Given that (X,τ) and (Y,τ) are homeomorphic, any topological property (such as connectedness) of (X,τ) or of (Y,τ) must be shared by both (X,τ) and (Y,τ).
• If (X,τ) and (Y,τ) do not share even a single topological property, then (X,τ) and (Y,τ) are not homeomorphic.     
• The common joke about how topologists consider donuts and coffee cups to be the same arises because those items are homeomorphic (if they are assumed to be topological spaces).

Hausdorff Spaces, Compactness, Paracompactness, and σ-Compactness

• Hausdorff spaces are topological spaces in which individual points have disjoint neighborhoods. Disjoint neighborhoods are neighborhoods that do not share any elements (the intersection of disjoint neighborhoods U and V is the null set).
• Note that the term “disjoint” can also refer to sets. Disjoint sets are sets that do not share any elements.

• Compact spaces have finite open covers. An open cover of a set X is a collection of sets that contains X as a subset. As such, a finite open cover involves a finite collection of sets containing X.
• Given a cover, a subcover of X is a subset of a cover that still contains X. The subcover of X is equivalent to the refinement of X.

• Paracompact topological spaces are spaces for which every open cover contains a locally finite open refinement. A collection of subsets is locally finite when each point in the space has a neighborhood that intersects a finite number of neighborhoods in the collection.
• σ-compact topological spaces are topological spaces which can be described as the union of countably many compact spaces.
• Note that a countable set is a set for which each element is associated with a unique natural number. More formally, countable sets have the same cardinality (number of elements) as a subset of the natural numbers. Countable sets may be countably finite or countably infinite. When countably infinite, each element can be counted one at a time, but the counting may never finish.

First-Countable and Second-Countable Spaces

• First-countable spaces are topological spaces for which every point is an element in a countable number of neighborhoods. For each point, this set of neighborhoods is called a neighborhood basis or a local basis.
• For example, consider a topological space on a set X consisting of four points x₁, x₂, x₃, x₄. (Assume that the set is equipped with a topology). Each point has a countable number of overlapping neighborhoods. In the figure, the neighborhoods which include point x₄ are shown as an example.

• All metric spaces are first-countable since any point in a metric space will have a countable (though often countably infinite) number of overlapping neighborhoods. This comes from the fact that distances are strictly defined on metric spaces, so the number of open balls making up the basis for a metric space will correspond to some natural number.
• Second-countable spaces are topological spaces with a countable basis.
• Euclidean space is a second-countable space because the set of open balls forming its basis can be restricted to the set of open balls with rational radii and centers with rational coordinates.
• Every second-countable space is first-countable (but not every first-countable space is second-countable).

Connectedness

• Connected spaces are topological spaces that cannot be represented as a union of disjoint nonempty open subsets. That is, a connected topological space X cannot be divided into disjoint nonempty open subsets.
• For instance, the usual topology (described in the next section) on Rⁿ is connected as any union of disjoint open subsets will always exclude at least part of the n-dimensional real numbers.
• Disconnected spaces are topological spaces which do not satisfy the definition of connected spaces.
• Path-connected spaces are topological spaces for which a path can be used to draw some curve from any point x to any point y in the given topological space X.
• A path is defined as a continuous function mapping from the unit interval [0,1] to the topological space X such that f(0) = x and f(1) = y.

• Every path-connected space is connected (but not every connected space is path-connected).
• Simply connected spaces are topological spaces which are path-connected and every path between every pair of points can be continuously transformed.
• Given a topological space V, it is said to be locally connected at a point x when every neighborhood of x contains a connected open neighborhood (a topological neighborhood that does not include its boundary). When V is locally connected at every point it contains, then V is called a locally connected topological space.

Essential Examples of Topological Spaces

• The standard (or usual) topology on Rⁿ may be defined as the union of all possible open balls on n-dimensional Euclidean space.
• Given a set X, the discrete topology on X is a topological space in which each point forms an open set. Each of these open sets is called a singleton (a set with only one element).
• Given a set X, the indiscrete (or trivial) topology is a topological space in which the only open sets are X itself and the empty set. If X has multiple elements, the space cannot be equipped with a metric (any distance between elements must be zero).
• Given a topological space (X,τ) and a subset V of X, the subspace topology is the subset V equipped with the topology τ.

Manifolds

• Topological manifolds are a type of topological space which must satisfy the conditions of Hausdorffness, second-countability, and paracompactness.
• In addition, topological manifolds must be locally homeomorphic to Euclidean space, Rⁿ.
• In order to be locally homeomorphic to Euclidean space, the formalisms below must hold.

• This means that, for every point x in a locally Euclidean topological space X, there exists an open set U such that x belongs to U and there exists a homeomorphism h from Rⁿ to U. These homeomorphisms are called charts. The combination of charts which covers X is called an atlas.
• Below is a simplified representation of a manifold (only a few homeomorphisms are shown). In this case, the manifold is a 2-manifold, also called a surface.

• Manifolds cannot “cross themselves” unless special conditions are met (i.e. the Klein bottle is an exception). For instance, a 1-manifold cannot have curves which intersect and a 2-manifold cannot pass part of its surface through another part of its surface. The reason for this is that Euclidean space cannot be homeomorphic to a point of intersection since this gives multiple values for the same function and therefore is not defined as a function.
• However, as the dimension of a manifold increases, the manifold can cross itself when projected into lower dimensions while not actually crossing itself in its highest dimension. Consider that a surface may “fold over” to some degree without intersecting itself in R³. If projected into R², then this manifold will appear to cross itself. But when looking back at the R³ depiction, it is clear that the manifold does not actually cross itself. The same holds true for higher dimensions.
• For an atlas, two charts can overlap on a manifold. The intersection of these two charts is an open set which maps to the same region of Euclidean space. Transition maps are composition functions f(g^-1) = f∘g^-1 and g(f^-1) = g∘f^-1 which map the open sets in R² to the manifold (by the inverse function) and then map back to Euclidean space (by the function which describes the other homeomorphism). This is also called a coordinate transformation.

• If a manifold’s transition maps are all at least once differentiable (C¹), then the manifold is called a differentiable manifold.
• If the transition maps are all infinitely differentiable (C^∞), then the manifold is called a smooth manifold.