Day 12/13/14/15/16/17/19 skipped 18 Topology

What is a limit point/ Add an image of this

A point x ∈ X is a limit point of A ⊆ X if for every open set U containing x, U contains a point of A other than x. A set A is closed iff it contains all of its limit points.

For any topological space X and subset A ⊆ X, we partition X into three subsets

- The interior of A is the subset Int A ⊆ A of all points x ∈ A such that there exists an open set U ⊆ X with x ∈ U ⊆ A - The boundary of A is the subset ∂A ⊆ X of all points x ∈ X such that, for any open set U ⊆ X containing x, U contains at least one point of A, and at least one point not in A The exterior of A is the subset Ext A ⊆ (X \ A) of all points x ∈ X \ A such that there exists an open set U ⊆ X with x ∈ U ⊆ (X \ A)

Closed sets X satisfy the following axioms

-Any union of finitely many closed sets is closed. - Any intersection of closed sets is closed. - ∅ and X are both closed

Ex of closed

For instance, a closed interval [a, b] is a closed set in the standard topology on R. In the discrete topology on X, every subset C ⊆ X is closed

Intuition abt limit points

Intuitively, limit points are points in X that are "infinitesimally close to A." One of the groups will prove that A is closed iff it contains all of its limit points. So, a set is closed when it contains all of the points that are infinitesimally close to it

Closed

Suppose X has a topology. A subset C ⊆ X is closed if the complement X \ C is open.

Closed

Suppose X has a topology. A subset C ⊆ X is closed if the complement X \ C is open. For instance, a closed interval [a, b] is closed in the standard topology on R

Continuity is local

We can check whether f : X → Y is continuous by checking it on one piece of X at a time.

Pasting Lemma

We can check whether f : X → Y is continuous by checking it on one piece of X at a time.

Homeomorphism

f is a homeomorphism if f is a bijection, and it is both continuous and open. In other words, both f and its inverse f^-1 are continuous. X and Y are homeomorphic if there exists a homeomorphism f : X → Y . We denote this by X ∼= Y .