# Topology

Topology is the study of mathematical objects endowed with the “smallest” structure necessary in order for continuity to be well-defined on said object.

Thus topology is, simply put, the study of continuity.

Highly relevant are also the properties preserved under general continuous deformations of topological spaces. These are called *topological properties*.

Some “basic” but very important examples are:

- Cardinality (dimension)
- Connectedness
- Compactness

This section from Wikipedia explains these very well:

dimension allows distinguishing between a line and a surface

compactness allows distinguishing between a line and a circle

connectedness allows distinguishing a circle from two non-intersecting circles

Topology, with its inherent abstractness and high level of rigour is often seen as a difficult subject. And it is often the first such subject students meet in university.

The reality is however that topology is in many ways quite visual and can lead to a deep understanding of geometry when properly understood and mastered.

## Topological spaces

As with many other objects under study in mathematics, topological spaces are sets endowed with particular structure.

In topology, this particular structure is called *a topology*.

A topology $\tau$ of a set $X$ is a is set of subsets of $X$. The topological space thus formed is denoted as $(X,\tau)$ or simply $X$ where the topology is taken from context.

The elements of $\tau$ (open sets) have to satisfy the following properties:

- $\empty$ and $X$ are in $\tau$
- Any union of some $A_i \in X$ is in $\tau$
- Any finite initersetion of some $A_i \in X$ is in $\tau$

### Open and closed sets

The members of a topology $\tau$ are called *open sets* and their complements are called closed.

$$ \textrm{A open} \iff A \in \tau $$ $$ \textrm{B closed} \iff B^C \in \tau $$

## Two basic topologies

### The discrete topology

$$\tau = \{A | A \sub X \}$$ In this topology all functions $f: X \to Y$ are continuous. This is the finest topology there is.

### The trivial topology

$$\tau = \{\empty, X\}$$ In this topology all functions $f: Y \to X$ are continuous. This is the coarsest topology there is.

## Continuous deformations

The deformations under study in topology are:

- homeomorphisms - two-way continuous transformations between topological spaces
- homotopies - two-way continuous transformations between homeomorphisms

## Interior, exterior, boundary and closure

### Interior points

In a metric space $(X,d)$ we say that $x_0 \in D \subset X$ is an interior point whenever:

$$\exist \epsilon > 0 \implies B_\epsilon(x_0) \subset D$$

(There is an open ball around the point fully contained in the subset)

This metric notion of interior poinits can be topologized by replacinig open balls with open sets.

### Boundary points

In a metric space $(X,d)$ we say that $x_0 \in X$ is a boundary point whenever:

$$\forall \epsilon > 0 \enspace \exist x,y \in B_\epsilon(x_0) \implies x \in D, y \in D^C$$

(Any open ball around the point is partially contained in ths space and in its complement.)

Equivalently, the condition for boundary points in a topological space are obtained by introducting open sets in place of open balls.

### Interior and boundary

$$ Int(D) = D\degree = \{x \in D | \textrm{x is interior point} \} $$ $$ Bndry(D) = \partial D = \{x \in D | \textrm{x is boundary point} \} $$

### Closure

The closure of a subset is the set itself together with its boundary:

$$ \bar{D} \equiv D \cup \partial D \ $$

### Links

## Big questions at a glance

Conceptual thoughts.

### What is a topology?

A topology is a choice of blurring between the elements of a set.

That is, it is a choice of which elements to group together and which elements one are to keep separate

### Why is a topology useful?

### What can a topology be used for?

### What properties are important in topology?

…

## Resources

- Introduction to Topology - Bert Mendelson