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 - Wikipedia

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:

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

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