---
lecture: 12
date: 2025-02-24
---
# Weakest Topology
Definition of [[Weakest Topology]] defined in the lecture.

# Initial Topology
Definition of [[Initial Topology]], defined in the lecture.
# Proposition
Let $X$ be a set with [[Initial Topology|initial topology]] induced by a family, $F$, of functions.

Then a [[Nets|net]] in $\{ x_{i} \}$ in $X$ converges to $x \iff f(x_{i}) \to f(x) \; \forall f \in F$.
## Proof
### $\Rightarrow$:
Is obvious.
### $\Leftarrow$:
Let $A$ be a neighbourhood of $x$.

Hence there are finitely many [[Open Sets|open sets]] $A_{n} \subset Y_{f_{n}}$ such that $x \in \cap f^{-1}_{n}(A_{n}) \subset A$. Then $A_{n}$ is a neighbourhood of $f_{n}(x), \; \forall n$.

By assumption $f_{n}(x_{i}) \to f_{n}(x)$, so $f_{n}(x_{i}) \in A_{n} \; \forall i \geq i(n)$ for some $i(n)$. Pick $j$ such that $j \geq i(n)$ for all the finitely many $n$s.

Then $x_{i} \in \cap f^{-1}_{n}(A_{n}) \subset A$ for all $i \geq j$, so $x_i \to x$.

QED.
# Corollary
$X$ has [[Initial Topology|initial topology]] induced by $F$. Say $Z$ is a [[Topological Space|topological space]], then:

$g : Z \to X$ is [[Continuous|continuous]] $f \circ g$ is [[Continuous|continuous]] $\forall f \in F$.
## Proof
### $\Rightarrow$:
Clear.
### $\Leftarrow$:
Say we have a [[Nets|net]] $\{ z_{i} \}$ that $z_{i} \to z$ in $Z$.

Then $\underbrace{(f \circ g)(z_{i})}_{= f(g(z_{i}))} \to \underbrace{(f \circ g)(z)}_{f(g(z))} \; \forall f \in F$.

Hence $g(z_{i}) \to g(z)$ by the proposition, so $g$ is [[Continuous|continuous]].
# Product Topology
Definition of the [[Product Topology]], defined in the lecture.

![[Drawing 2025-02-24 11.58.37.excalidraw.dark.svg]]
%%[[Drawing 2025-02-24 11.58.37.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-02-24 11.58.37.excalidraw.light.svg|light exported image]]%%

$(x_{i}, y_{i}) \to (x,y)$

By the previous proposition a net $\{ x_{i} \}$ in $\Pi X_{\lambda}$ converges to $x$ with respect to the [[Product Topology|product topology]] $\iff \pi_{\lambda} \to \pi_{\lambda}(x), \; \forall \lambda$.
### Note
$\pi_{\lambda}$ are [[Continuous|continuous]] (obvious) and [[Open Sets|open sets]] $\pi_{\lambda}(A)$ open in $X_{\lambda}$ for $A$ open in $\Pi X_{\lambda}$
## Tychonoff
It is the [[Tychonoff Theorem]] (defined in lecture).
# Separating Family
Defines [[Separating Points]] from the lecture.
# Proposition
A set $X$ with [[Initial Topology|initial topology]] induced from a [[Separating Points|separating family]] of functions $f : X \to Y_{f}$ is [[Hausdorff]] when all $Y_{f}$ are [[Hausdorff]].
## Proof
Say $x \neq y$ in $X$. Then $\exists f$ such that $f(x) \neq f(y)$ in $Y_{f}$.

Can separate $f(x)$ and $f(y)$ by neighbourhoods $U$ and $V$ such that $U \cap V = \emptyset$.

Then $f^{-1}(U)$ and $f^{-1}(V)$ will be disjoint neighbourhoods of $x$ and $y$.

> [!note] Reasoning for $f^{-1}(U)$ and $f^{-1}(V)$ being disjoint
> $$z \in f^{-1}(U) \cap f^{-1}(V)$$
> $$f(z) \in U \cap f(z) \in V$$
> Which is not possible

QED.
# Corollary
A product of [[Hausdorff]] spaces is [[Hausdorff]] in the [[Product Topology|product topology]].

$\Pi X_{\lambda}$
$\Pi_{\lambda}$

$f: X \to Y_{f}$
$f : X_{f} \to Y$

$q : X \to X \setminus \sim$
$q(x) = [x]$