General (point-set) topology Jundong Liu Ohio Univ.

Motivations (ε, δ)-definition of limit & continuity – For (ε, δ), we need a definition of distance Topology is the study of shape without distance From metric spaces to topological spaces – Generalization of “nearness” from (ε, δ) to neighborhoods – Open sets for neighborhoods ( )

Properties of neighborhoods Each point should belong to its own neighborhoods Union of two neighborhoods should still be a neighborhood Intersection of two neighborhoods should still be a neighborhood

Use open set to represent neighborhood Open intervals in R1 (real line) Open balls in R2 (real plane) Generalize open intervals and open balls to open sets in metric space – Definition: an open set, U, is any subset of X such that, for any point p in U, there exists an open ball (well defined in metric a space) at p which is entirely contained in U.

Further generalize the concept of open set the empty set is open a finite intersection of open sets is open an arbitrary (i.e. possibly infinite) union of open sets is open These three axioms become the defining properties of open sets in topological spaces. An open set about (containing) a particular element (point) of the space is called a neighborhood of that point.

Topology T on a set X Given a set X, a collection τ of subsets of X can be called a “ (open-set) topology on X” if and only if – X (the whole set) is an element of τ – The empty set is an element of τ – For any two elements in τ, their intersection is in τ – For any two elements in τ, their union is in τ (X, τ) is called a topological space. Each element is a open set – a neighborhood

Closed set A set C is closed in the topology if and only if X – C \in τ – The empty set and X are closed. – The intersection of any collection of closed sets is also closed. – The union of any pair of closed sets is also closed. Topology can also be defined through closed sets.

Examples τ

Examples of topologies on X = {1, 2, 3, 4} trivial topology (indiscrete topology). – X = {1, 2, 3, 4} and collection τ = {{}, {1, 2, 3, 4}} discrete topology – X = {1, 2, 3, 4} and collection τ = P(X) (the power set of X) X = {1, 2, 3, 4} and collection τ = {{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}, {1, 2, 3, 4}} of six subsets of X form another topology.

Examples X = Z, the set of integers, and collection τ equal to all finite subsets of the integers plus Z itself is not a topology, because (for example) the union of all finite sets not containing zero is infinite but is not all of Z, and so is not in τ.

Example 10 apples and 5 oranges

A lot of more from wiki: R There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set.

Rn and Cn More generally, the Euclidean spaces Rn can be given a topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complex numbers, and Cn have a standard topology in which the basic open sets are open balls.

Metric spaces Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

Set of linear operators Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

Topology on manifold Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn.

Generalized concepts in topological spaces Continuity Homeomorphisms Topological invariants – Connectedness – Compactness

Continuity From (ε, δ) to open sets ( pological_spaces ) pological_spaces A function f: X->Y between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

Homeomorphisms When would we consider {X, T_x} and {Y, T_y} to be the same topological spaces? We want correspondence at the level of open sets. – f maps open sets in X to open sets in Y, and f^-1 maps open sets in Y to X. – This implies that both f and f^-1 are continuous.