The Eigenspace

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/Part 2: General Topology/Chapter 3: Topological Structure

3.1 Generating Topologies

Definition 3.1.1 (Topological Space).label Let $X$ be a set, possibly empty. A topology on $X$ is a structure given by a collection $\tau\suf\mathcal{P}(X)$ satisfying:

  1. (O1)

    $\emp,X\in \tau$

  2. (O2)

    $\fall \curl{U_i}_{i\in I}\suf \tau,\cups{}{i\in I}U_{i}\in \tau$

  3. (O3)

    $\fall U,V\in \tau, U\cap V\in \tau$

The ordered pair $(X,\tau)$ (or just $X$ if $\tau$ is understood) is a topological space and elements of $\tau$ are open sets.

The elements of a topological space are points and $X$ is the underlying set of the topological space $(X,\tau)$. If $\tau\suf \tau'$ are topologies on $X$ then $\tau$ is courser than $\tau'$ and $\tau'$ is finer than $\tau$.

Definition 3.1.2 (Closed Set).label Let $(X,\tau)$ be a topological space, then $A\suf X$ is closed if $A^{c}\in\tau$.

Definition 3.1.3 (Seperation Axioms).label Let $(X,\tau)$ be a topological space, then we have the following separation axioms which $X$ might satisfy

  1. (T0)

    For any $x\neq y\in X$ there exists $U\in \tau$ such that either $y\nin U\ni x$ or $x\nin U\ni y$

  2. (T1)

    For any $x\neq y\in X$ there exists $U\in\tau$ such that $y\nin U\ni x$

  3. (T2)

    For any $x\neq y\in X$ there exists $U,V\in\tau$ such that $x\in U,y\in V$ and $U\cap V=\emp$

  4. (T3)

    $X$ is (T1) and for any $x\in X,A\suf X$ closed with $x\nin A$ there exists $U,V\in\tau$ such that $x\in U,A\suf V$ and $U\cap V=\emp$

  5. (T4)

    $X$ is (T1) and for any $A,B\suf X$ closed with $A\cap B=\emp$ there exists $U,V\in\tau$ such that $A\in U,B\suf V$ and $U\cap V=\emp$

Definition 3.1.4 (Base of Topology).label Let $X$ be a set then $\cali{B}\suf\cali{P}(X)$ is a base if it satisfies:

  1. (B1)

    For each $x\in X$ there exists $B_{x}\in\cali{B}$ such that $x\in B_{x}$

  2. (B2)

    If $x\in B_{1},B_{2}\in\cali{B}$ then there exists $B_{3}\in\cali{B}$ such that $x\in B_{3}\suf B_{1}\cap B_{2}$

then the topology generated by $\cali{B}$ defined as
\begin{align*}\tau(\cali{B})\define\curl{\cups{}{i\in I}U_i:\curl{U_i}_{i\in I}\suf \cali{B},I\text{ index set}}\end{align*}

is a topology on $X$. Suppose $X$ has a topology $\tau$ then we say $\cali{B}$ is a base for $\tau$ if $\cali{B}\suf \tau$ and

  1. (B’1)

    $\cali{B}$ satisfies (B1)

  2. (B’2)

    For each $x\in X, U\in\tau$ such that $x\in U$, there exists $B\in\cali{B}$ such that $x\in B\suf U$

in which case $\tau=\tau(\cali{B})$.

Proof. If $\cali{B}$ is a base then $\emp=\cups{}{\emp}\in\tau(\cali{B})$ and by (B1) $X=\cups{}{x\in X}B_{x}\in \tau(\cali{B})$ showing (O1). Observe that for any $\curl{U_i}_{i\in I},\curl{V_j}_{j\in J}\suf\cali{B}$ and $*\in \curl{\cap,\cup}$ we have

\begin{align*}\parens{\cups{}{i\in I}U_i}*\parens{\cups{}{j\in J }V_j}=\cups{}{i\in I}\cups{}{j\in J}U_{i}* V_{j}\end{align*}

which shows (O2) and by (B2) this observation also shows (O3).

Suppose now that $\cali{B}$ is a base for $\tau$ then by (O3) and $\cali{B}\suf \tau$ we have $\tau(\cali{B})\suf \tau$. For any $U\in \tau$ either $U=\emp\in\tau(\cali{B})$, or $U\neq \emp$. In which case by (B’2) for each $x\in U$ pick $B_{x}\in\cali{B}$ such that $x\in B_{x}\suf U$ then $U=\cups{}{x\in U }B_{x}\in \tau(\cali{B})$ as desired.$\square$

Definition 3.1.5 (Covering).label Let $A\suf X$ be subset of a topological space. A collection $(U_{i})_{i\in I}$ is called a covering of $A$ if $A\suf\cups{}{i\in I}U_{i}$ and $U_{i}\suf X$ for each $i\in I$. The covering is said to be open if all $U_{i}$ are open in $X$.

The Eigenspace

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