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 coarser 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 $A^{c}\in\tau$. We say $A\suf X$ is clopen if $A$ is open and closed.

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 $A\suf U,B\suf V$ and $U\cap V=\emp$

Definition 3.1.4 (Covering).label Let $X$ be a set and $A\suf X$. A collection $\curl{U_i}_{i\in I}$ is called a covering of $A$ if $A\suf\cups{}{i\in I}U_{i}$ and $U_{i}\suf X$ for each $i\in I$. A covering $\curl{U_i}_{i\in I}$ is said to be open (resp. closed) if $X$ is a topological space and all $U_{i}$ are open (resp. closed) in $X$. A covering $\curl{U_i}_{i\in I}$ is said to be (pairwise)disjoint if for each $i,j\in I$, $U_{i}\cap U_{j}=\begin{cases}\emp&i\neq j\\ U_{i}&i=j\end{cases}$. A subcover of a covering $\curl{U_i}_{i\in I}$ is a covering $\curl{V_j}_{j\in J}$ such that $J\suf I$ and $V_{j}=U_{j}$ for each $j\in J$ and $\curl{V_j}_{j\in J}$ is said to be finite if $J$ is finite.

Definition 3.1.5 (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 $x\in B_{x}$ i.e. $\cali{B}$ is a covering of $X$

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

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 $\tau=\tau(\cali{B})$.

Definition 3.1.6 (Subbase of Topology).label Let $X$ be a set then $\cali{S}\suf\cali{P}(X)$ is a subbase if it is a covering of $X$. The smallest topology on $X$ containing $\cali{S}$ is given by

where $\tau(\cali{S})$ is called the topology generated by $\cali{S}$ and

is a base for $\tau(\cali{S})$.

Definition 3.1.7 (Intial Topology).label Let $X$ be a set, $\curl{(Y_i,\tau_i)}_{i\in I}$ be a family of topological spaces, and $\curl{f_i:X\to Y_i}_{i\in I}$ be a family of maps then there exists a topology $\tau$ on $X$ satisfying:

1. (1)

   $f_{i}\in C(X;Y_{i})$ for each $i\in I$

2. (U)

   $\tau$ is the coarsest topology on $X$ such that all $f_{i}$ are continuous

3. (U’)

   For any topological space $Z$ and map $g:Z\to X$ we have $g$ is continuous if and only if $f_{i}\circ g:Z\to Y_{i}$ is continuous for each $i\in I$.

4. (4)

   A subbase for $\tau$ is given by

   \begin{align*}\cali{S}=\curl{f_i\inv(U_i):U_i\in\tau_i,i\in i}\end{align*}

   
   
5. (5)

   (1)+(U)$\iff$(U’)

called the intial/weak topology on $X$ generated by the maps $\curl{f_i}_{i\in I}$.

Definition 3.1.8 (Final Topology).label Let $X$ be a set, $\curl{(Y_i,\tau_i)}_{i\in I}$ be a family of topological spaces, and $\curl{f_i:Y_i\to X}_{i\in I}$ be a family of maps then there exists a topology $\tau$ on $X$ satisfying:

1. (1)

   $f_{i}\in C(Y_{i};X)$ for each $i\in I$

2. (U)

   $\tau$ is the finest topology on $X$ such that all $f_{i}$ are continuous

3. (U’)

   For any topological space $Z$ and map $g:X\to Z$ we have $g$ is continuous if and only if $g\circ f_{i}:Y_{i}\to Z$ is continuous for each $i\in I$.

4. (4)

   (1)+(U)$\iff$(U’)

called the final/strong topology on $X$ generated by the maps $\curl{f_i}_{i\in I}$.

Definition 3.1.9 (Generated Topology).label Let $X$ be a topological space and $\s\suf\cali{P}(X)$ then $X$ is $\s$-generated if the topology on $X$ is the final topology generated by the inclusion maps $\curl{\iota_S:S\to X:S\in\s}$. If $\kappa\suf \cali{P}(X)$ is the collection of precompact sets of $X$ and $X$ is $\kappa$-generated then $X$ is compactly generated.

Definition 3.1.10 (Subspace Topology).label Let $(X,\tau_{X})$ be a topological space and $A\suf X$ be a subset then the subspace topology on $A$

is the initial topology on $A$ with respect to the inclusion map $\iota:A\inj X$.