Topology

Here are some of the most basic definitions from topology.

Definition 1

  A set X is called a topological space  with topology ${\cal{T}}$, provided there exists a family ${\cal{T}}$ of subsets X satisfying:

• $X\in {\cal{T}}$ og $\emptyset \in{\cal{T}}$.
• $O_1,\dots,O_r\in{\cal{T}}\Rightarrow O_1\cap\dots\capO_r\in {\cal{T}}$.
• If $\{O_\alpha\}_{\alpha\in A}\subseteq {\cal{T}}$, then $\cup_{\alpha\in A}O_\alpha\in {\cal{T}}$.

The elements in ${\cal{T}}$ are called the open subsets.

If $x\in X$ a neighborhood of x is by definition an open subset O, such that $x\in O$.

The space X is called a Hausdorff space if

$$\forall x_1\neq x_2\in X: \exists O_1,O_2\in {\cal{T}}:x_1\in O_1,x_2\in O_2\text{ og }O_1\cap O_2=\emptyset .$$ (1)

A set X can always be made into a topological space by letting${\cal{T}}=\{X,\emptyset \}$. This is called the trivial topology. In the opposite extreme, one may choose ${\cal{T}}$ as the family of all subsets. This is not good for much either. We will write $(X,{\cal{T}})$ if we are in the situation of Definition 1.

Definition 2

Let $(X_1,{\cal{T}}_1)$ og $(X_2,{\cal{T}}_2)$ be topological spaces. A map $f:X_1\rightarrow X_2$ is said to be continuous (with respect to the considered topologies) provided

$$\forall U\in {\cal{T}}_2:f^{-1}(U)\in {\cal{T}}_1.$$ (2)

Definition 3

En sub family $\cal{B}\subseteq {\cal{T}}$ is called a basis for the topology  or a basis for the open subsets, if any $O\in{\cal{T}}$ can be written as the union of (some) sets from $\cal{B}$. The space X is said to be second countable if there exists a countable basis of the open sets.

Definition 4

A topological space $(X,{\cal{T}})$ is disconnected, provided there exist two open non-empty sets O₁,O₂ af X such that

$$X=O_1\cup O_2\text{ og}O_1\cap O_2=\emptyset .$$ (3)

If X is not disconnected, X is called connected .

Definition 5

If $(X,{\cal{T}})$ is a topological space and $Y\subset X$,

$${\cal{T}}_Y=\{O\cap Y\mid O\in {\cal{T}}\}$$ (4)

defines a topology on Y called the induced topology .

Remark 6

For a regular surface S in ${\Bbb R}^3$ the induced topology is exactly the topology one obtains by viewing S as a metric space, where the metric is inherited from the metric space ${\Bbb R}^3$. As a basis for the topology one may thus choose the sets

$$K_S(s,r)=S\cap K(s,r)=\{s'\in S\mid \Vert s-s'\Vert<r\}$$ (5)

for $s\in S$ and r>0.