Point Set Topology

Order/Simple Order Relation Definition

Axiom of Choice

Well-Ordered Definition

(least w.r.t. the particular order relation)

Well-Ordering Theorem

Section Definition

Strict Partial Order Definition

The Maximum Principle Theorem

There exists a well-ordered set A having a largest element… Lemma

If A is a countable subset of S_omega…

Partial Order Definition

Topology and Topological Space Definitions

Open Sets Definition

Discrete Topology

Indiscrete Topology

Finer vs Coarser (and strict)

Comparable Definition

Topology top left is coarser than the others since it is contained in all of them. Topology top right is finer than the left column and coarser than the two below it. It is not comparable to those in the middle column.

Basis for a Topology Definition

Topology generated by Basis B Definition

T equals the collection of all unions… Lemma 13.1

Suppose C is a collection of open sets of X such that for each open set U of X and each x in U… Lemma 13.2

One topology finer than another (on X) is equivalent to… Lemma 13.3

Standard Topology Definition

Lower Limit Topology on R Definition

K-Topology on R Definition

Rl and Rk are finer than.. Lemma 13.4

Subbasis (and Topology generated by it) Definitions

Order Topology Definition

Rays (open and closed) Definitions

Product Topology Definition

Product Topology Smaller Basis Theorem

Projections Definition

Inverse Projections being a subbasis for.. Theorem

Subspace Topology Definition

Basis for the subspace topology on Y.. Lemma 16.1

For a subspace of X, if U is open in Y and Y is open in X…

Subspace Topology and Product Topology Theorem

Order Topology and Subspace Topology Theorem 16.4

Conditions for closed sets of a topological space Theorem

Let Y be a subspace of X. Then a set A is closed in Y iff.. Theorem

Let Y be subspace of X, if U is closed in Y and Y is closed in X…

Interior and Closure Definition

Let Y be a subspace of X, let A be a subset of Y, Then the closure of A in Y is… Theorem

If A is a subset of X, then x is in A closure iff..

Limit Point Definition

The closure of a set A equals the union of… Theorem

A subset of a Topological space is closed iff…

Converge Definition

Hausdorff Space Definition

Every finite point set in a Hausdorff Space…

Every finite point set being closed is the T1 axiom

If X satisfies T1 axiom, then x is a limit point of the subset A iff…Theorem

If X is a Hausdorff Space then sequence of points of X… Theorem

Simply ordered sets, product of… Subspace of.. Theorem

Continuity and Preimage Definitions

Equivalent to f being continuous (3 other things) Theorem

Homeomorphism Definition

Definition of Imbedding

Rules for constructing continuous functions Theorem

The Pasting Lemma

Maps into Products Theorem

J-tuple Definition

Cartesian Product Definition

Box Topology Definition

Projection Mapping Definition

subbasis of the product topology and product space definitions

Comparison of the box and product topologies Theorem

bases for the box and product topologies Theorem

cartesian products and Subspaces, Hausdorff spaces, cartesian product of the closure of a collection equals… Theorems

function mapped to cartesian product and continuity Theorem

Metric Definition

Metric Topology Definition

Metrizable Definition

Bounded and Diameter definitions

Standard bounded metric Definition/Theorem

norm, euclidean metric, square metric Definitions

Metric topologies and who is finer.. Theorem

euclidean metric topology and square metric topology are the same as… Theorem

uniform metric and topology Definition

uniform topology vs product topology vs box topology thm

metric that induces the product topology on Rw… Theorem

Let f: X → Y where X and Y are metrizable with different metrics. Then the continuity of f is equivalent to… Theorem

The sequence lemma

If f: X → Y is continuous, then for every convergent sequence x_n → x in X… converse? Theorem

If X is a topological space and if f,g: X → R are continuous…. Theorem

Uniform Convergence Definition

Uniform Limit Theorem

Quotient Map Definition

Saturated Definition

Open and Closed Maps Definitions

Quotient Topology Definition

Quotient Space

What do you need, to ensure the restriction of a quotient map is itself a quotient map? Theorem

Induced map from a quotient map and g Theorem

Let g : X → Z be a surjective continuous map. Let X* be the following collection of subsets of X: … give X* the quotient topology. Then… Corollary

Separation and Connected Definitions

If Y is a subspace of X, a separation of Y is… Theorem

5 easy results on Connectedness

proof of last one

Linear Continuum Definition

If L is a linear continuum in the order topology… Theorem

+ corollary