Inverse-producing extensions of Topological Algebras/topological algebra

Definition: Topological Vector Space

A topological vector space $V$ over $𝕂$ is a vector space over the field $𝕂$ that has a topology with which scalar multiplication and addition are continuous mappings.

$\begin{array}{ccc}\cdot :𝕂\times V& ⟶& V(\lambda ,v)⟼\lambda \cdot v\\ +:V\times V& ⟶& V(v,w)⟼v+w\end{array}$

In the following, for all topological vector spaces, we shall use the Hausdorff property be assumed.

Definition: Neighbourhood

Let $(X,𝒯)$ be a topological space with a topology $𝒯$ as a system of open sets $𝒯\subset \mathrm{\wp }(X)$ and $a\in X$, then denote

• ${𝔘}_{𝒯}(a):=\{U\subseteq X:{\mathrm{\exists }}_{U_o\in 𝒯}:a\in U_o\subseteq U\}$ the set of all neighbourhoods from the point $a$,
• ${\stackrel{o}{𝔘}}_{𝒯}(a):={𝔘}_{𝒯}(a)\cap 𝒯$ the set of all open Neighbourhoods from the point $a$,
• ${\overline{𝔘}}_{𝒯}(a):=\{\bar{U}:U\in {𝔘}_{𝒯}(a)\}$ the set of all closed neighbourhoods of point $a$.

Remark: Indexing with topology

If no misunderstanding about the underlying topological space can occur, the index $𝒯$ is not included as a designation of the topology used.

Remark: Analogy to the epsilon neighbourhood

In convergence statements in the real numbers one usually considers only $\epsilon$ neighbourhood. In doing so, one would actually have to consider in topological spaces for arbitrary neighbourhoods from $U\in {𝔘}_{𝒯}(a)$ find an index bound $i_U\in I$ of a net $(x_i{)}_{i\in I}$ above which all $x_i\in U$ lie with $i\ge i_U$. However, since the $\epsilon$ neighbourhoods are an neighbourhood basis, by the convergence definition one only needs to show the property for all neighbourhoods with $\epsilon >0$.

Convergence in topological spaces

Let $(X,𝒯)$ be a topological space, $a\in X$, $I$ an index set (partial order) and $(x_i{)}_{i\in I}\in X^I$ a mesh. The convergence of $(x_i{)}_{i\in I}\in X^I$ against $a\in X$ is then defined as follows:

$$\begin{array}{ccc}\stackrel{𝒯}{\underset{i\in I}{\lim }}{x}_i=a& :⟷& {\mathrm{\forall }}_{U\in {𝔘}_{𝒯}(a)}{\mathrm{\exists }}_{i_U\in I}{\mathrm{\forall }}_{i\ge i_U}:x_i\in U\end{array}$$.

(where "$\le$" for $I$ is the partial order on the index set).

Definiton: Neighbourhood basis

Let $(X,𝒯)$ be a topological space, $a\in X$ and ${𝔘}_{𝒯}(a)$ the set of all neighbourhoods of $a\in X$. ${𝔅}_{𝒯}(a)$ is called the neighbourhood basis of ${𝔘}_{𝒯}(a)$ if for every :${𝔅}_{𝒯}(a)\subseteq {𝔘}_{𝒯}(a)\wedge {\mathrm{\forall }}_{U\in {𝔘}_{𝒯}(a)}{\mathrm{\exists }}_{B\in {𝔅}_{𝒯}(a)}:B\subseteq U$.

Remark: Epsilon spheres in normalized spaces

Let $(V,\Vert \cdot \Vert )$ be a normed space, then the $\epsilon$ spheres form

$B_{\epsilon }^{\Vert \cdot \Vert }(a):=\{v\in V;\Vert v-a\Vert <\epsilon \}$

an ambient basis of ${𝔅}_{𝒯}(a)$ the set of all environments of ${𝔘}_{𝒯}(a)$ of $a\in V$.

Learning Task 1

Let $(X,𝒯)$ be a toplogic space with chaotic topology $𝒯:=\{\mathrm{\varnothing },X\}$.

• Determine ${𝔘}_{𝒯}(a)$ for any $a\in X$.
• Show that any sequence $(x_n{)}_{n\in \mathbb{N}}\in X^{\mathbb{N}}$ converges in $(X,𝒯)$ against any limit $a\in X$.

Learning Task 2

Let $(X,d)$ be a metric space with the discrete topology given by the metric:

$d(x,y):=\{\begin{array}{ccc}0& \text{ for }& x=y\\ 1& \text{ for }& x=y\end{array}$.

• Determine ${𝔘}_{𝒯}(a)$ for any $a\in X$.
• How many sets make up ${𝔅}_{𝒯}(a)$ minimal for any $a\in X$?
• Formally state all sequences $(x_n{)}_{n\in \mathbb{N}}\in X^{\mathbb{N}}$ in $(X,d)$ that converge to a limit $a\in X$!

Definition: open sets

Let $(X,𝒯)$ be a topological space and $𝒯\subseteq \mathrm{\wp }(X)$ be the system of open sets, that is:

$U\subseteq X\text{ open }:⟺U\in 𝒯$.

Task

Let $(ℝ,𝒯)$ be a topological space on the basic set of real numbers. However, the topology does not correspond to the Euclic topology over the set $|\cdot |$, but the open sets are defined as follows.

$U\in 𝒯\text{ open }:⟺U=\mathrm{\varnothing }\text{ or }U\subseteq ℝ\text{ with }{U}^c\text{ countable}$

• Show that $(ℝ,𝒯)$ is a topological space.
• Show that the sequence ${\left(\frac{1}{n}\right)}_{n\in \mathbb{N}}$ does not converge to $0$ in the topological space $(ℝ,𝒯)$.

Here $U^c:=ℝ\setminus U$ is the complement of $U$ in $ℝ$.

Remark: open - closed

By the system of open sets in a topology $𝒯\subseteq \mathrm{\wp }(X)$ the closed sets of the topology are also defined at the same time as their complements.

Definition: closed sets

Let $(X,𝒯)$ be a topological space and $𝒯\subseteq \mathrm{\wp }(X)$ be the system of open sets.

$M\subseteq X\text{ completed }:⟺{\mathrm{\exists }}_{U\in 𝒯}:M=U^c:=X\setminus U$

Definition: open kernel

Let $(V,𝒯)$ be a topological space and $M\subset V$, then the open kernel $\stackrel{\circ }{M}$ of $M$ is the union of all open subsets of $M$.

$$\stackrel{\circ }{M}:=\bigcup _{U\in 𝒯,U\subseteq M}U$$.

Definition: closed hull

Let $(X,𝒯)$ be a topological space. The closed hull $\bar{M}$ of $M$ is the intersection over all closed subsets of $W=U^c$ containing $M$ and $U$ is open.

$$\bar{M}:=\bigcap _{U\in 𝒯,U^c:=X\setminus U\supseteq M}{U}^c$$

Definition: edge of a set

The topological edge $\partial M$ of $M$ is defined as follows:

$$\partial M:=\bar{M}\mathrm{\setminus }\stackrel{\circ }{M}$$

Remark: sequences and nets

In metric spaces, one can still work with the natural numbers as countable index sets. In arbitrary topological spaces one has to generalize the notion of sequences to the notion of nets.

Definition: nets

Let $T$ be a topological space and $I$ an index set (with partial order), then $T^I$ denotes the set of all families indexed by $I$ in $T$:

$$T^I:=\{(t_i{)}_{i\in I}:t_i\in T\text{ for all }i\in I\}$$

Definition: finite sequences

Let $V$ be a vector space, then $c_{oo}(V)$ denotes the set of all finite sequences with elements in $V$:

$$c_{oo}(V):=\{(v_n{)}_{n\in {\mathbb{N}}_0}\in V^{{\mathbb{N}}_0}:{\mathrm{\exists }}_{N\in {\mathbb{N}}_0}{\mathrm{\forall }}_{n\ge N}:v_n=0\}.$$

Definition: Algebra

An algebra $A$ over the field $𝕂$ is a vector space over $𝕂$ in which a multiplication is an inner join

$$\cdot :A\times A⟶A(v,w)⟼v\cdot w$$

is defined where for all $x,y,z\in A$ and $\lambda \in 𝕂$ the following properties are satisfied:

$\begin{array}{ccc}x\cdot (y\cdot z)& =& (x\cdot y)\cdot z\\ .x\cdot (y+z)& =& x\cdot y+x\cdot z\\ (x+y)\cdot z& =& x\cdot z+y\cdot z\\ \lambda \cdot (x\cdot y)& =& (\lambda \cdot x)\cdot y=x\cdot (\lambda \cdot y)\end{array}$

Definition: topological algebra

A topological algebra $(A,{𝒯}_A)$ over the field $𝕂$ is a topological vector space $(A,{𝒯}_A)$ over $𝕂$, where also multiplication is

$$\cdot :A\times A⟶A(v,w)⟼v\cdot w$$

is a continuous inner knotting.

Continuity of multiplication

Continuity of multiplication means here:

$${\mathrm{\forall }}_{U\in 𝔘(0)}{\mathrm{\exists }}_{V\in 𝔘(0)}:V\cdot V=V^2\subset U$$

Multiplicative topology - continuity

The topology is called multiplicative if holds:

$${\mathrm{\forall }}_{U\in 𝔘(0)}{\mathrm{\exists }}_{V\in 𝔘(0)}:V^2\subset V\subset U$$

Remark: Multiplicative topology - Gaugefunctionals

In describing topology, the Topologization Lemma for Algebras shows that the topology can also be described by a system of Gaugefunctionals

Unitary algebra

The algebra $A$ is called unital if it has a neutral element $e$ of multiplication. In particular, one defines $x^o:=e$ for all $x\in A$. The set of all invertible (regular) elements is denoted by $𝒢(A)$. Non-invertible elements are called singular.

Task: matrix algebras

Consider the set $V$ of square $2\times 2$ matrices with matrix multiplication and the maxmum norm of the components of the matrix. Try to prove individual properties of an algebra ($V$ is a non-commutative unitary algebra). For the proof that $V$ with matrix multiplication is also a topological algebra, see Topologization Lemma for Algebras.

Definition: sets and links

Let $(A,𝒯)$ be a topological algebra over the field $𝕂$, $\mathrm{\Lambda }\subset 𝕂$ and $M_1,M_2$ be subsets of $A$, then define

$\begin{array}{ccc}{M}_1\times M_2& :=& \{(m_1,m_2)\in A\times A:m_1\in M_1\wedge m_2\in M_2\}\\ M_1+M_2& :=& \{m_1+m_2:m_1\in M_1\wedge m_2\in M_2\}\\ M_1\cdot M_2& :=& \{m_1\cdot m_2:m_1\in M_1\wedge m_2\in M_2\}\\ \mathrm{\Lambda }\cdot M_1& .=& \{\lambda \cdot m_1:m_1\in M_1\wedge \lambda \in \mathrm{\Lambda }\}.\end{array}$

Learning Tasks

Draw the following set $M_k$ of vectors as sets of points in the Cartesian coordinate system ${ℝ}^2$ with $M_1:=\{\left(\begin{array}{c}12\end{array}\right),\left(\begin{array}{c}10\end{array}\right)\}$ and $M_2:=\{\left(\begin{array}{c}32\end{array}\right),\left(\begin{array}{c}01\end{array}\right)\}$ and the following intervals $[a,b]\in ℝ$:

• $[1,4]\times [2,3]$.
• $M_1+M_2$.
• $[1,2]\cdot M_1$.

See also

• Topological vector space

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