From 000922f9bd2e265e62ead9f745ffb65deaa92782 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Sat, 22 Nov 2025 15:18:52 -0600 Subject: [PATCH] bugfix --- content/CSE5313/CSE5313_L15.md | 2 + content/CSE5313/CSE5313_L18.md | 12 +-- content/Math4201/Math4201_L35.md | 125 +++++++++++++++++++++++++++++++ content/Math4201/_meta.js | 1 + 4 files changed, 134 insertions(+), 6 deletions(-) create mode 100644 content/Math4201/Math4201_L35.md diff --git a/content/CSE5313/CSE5313_L15.md b/content/CSE5313/CSE5313_L15.md index cf050e2..d3dfb58 100644 --- a/content/CSE5313/CSE5313_L15.md +++ b/content/CSE5313/CSE5313_L15.md @@ -140,6 +140,7 @@ $$ \begin{aligned} H(Y|X=x)&=-\sum_{y\in \mathcal{Y}} \log_2 \frac{1}{Pr(Y=y|X=x)} \\ &=-\sum_{y\in \mathcal{Y}} Pr(Y=y|X=x) \log_2 Pr(Y=y|X=x) \\ +\end{aligned} $$ The conditional entropy $H(Y|X)$ is defined as: @@ -150,6 +151,7 @@ H(Y|X)&=\mathbb{E}_{x\sim X}[H(Y|X=x)] \\ &=-\sum_{x\in \mathcal{X}} Pr(X=x)H(Y|X=x) \\ &=-\sum_{x\in \mathcal{X}, y\in \mathcal{Y}} Pr(X=x, Y=y) \log_2 Pr(Y=y|X=x) \\ &=-\sum_{x\in \mathcal{X}, y\in \mathcal{Y}} Pr(x)\sum_{y\in \mathcal{Y}} Pr(Y=y|X=x) \log_2 Pr(Y=y|X=x) \\ +\end{aligned} $$ Notes: diff --git a/content/CSE5313/CSE5313_L18.md b/content/CSE5313/CSE5313_L18.md index f76508a..a7a87b0 100644 --- a/content/CSE5313/CSE5313_L18.md +++ b/content/CSE5313/CSE5313_L18.md @@ -196,7 +196,7 @@ $\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(U_1, Conclude similarly by the law of total probability. -$\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(s_\mathcal{Z}) \implies I(S_\mathcal{Z}; M_1, \cdots, M_{t-z}) = 0. +$\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(s_\mathcal{Z}) \implies I(S_\mathcal{Z}; M_1, \cdots, M_{t-z}) = 0$. ### Conditional mutual information @@ -246,14 +246,14 @@ A: Fix any $\mathcal{T} = \{i_1, \cdots, i_t\} \subseteq [n]$ of size $t$, and l $$ \begin{aligned} H(M) &= I(M; S_\mathcal{T}) + H(M|S_\mathcal{T}) \text{(by def. of mutual information)}\\ -&= I(M; S_\mathcal{T}) \text{(since S_\mathcal{T} suffice to decode M)}\\ -&= I(M; S_{i_t}, S_\mathcal{Z}) \text{(since S_\mathcal{T} = S_\mathcal{Z} ∪ S_{i_t})}\\ +&= I(M; S_\mathcal{T}) \text{(since }S_\mathcal{T}\text{ suffice to decode M)}\\ +&= I(M; S_{i_t}, S_\mathcal{Z}) \text{(since }S_\mathcal{T} = S_\mathcal{Z} ∪ S_{i_t})\\ &= I(M; S_{i_t}|S_\mathcal{Z}) + I(M; S_\mathcal{Z}) \text{(chain rule)}\\ -&= I(M; S_{i_t}|S_\mathcal{Z}) \text{(since \mathcal{Z} ≤ z, it reveals nothing about M)}\\ +&= I(M; S_{i_t}|S_\mathcal{Z}) \text{(since }\mathcal{Z}\leq z \text{, it reveals nothing about M)}\\ &= I(S_{i_t}; M|S_\mathcal{Z}) \text{(symmetry of mutual information)}\\ &= H(S_{i_t}|S_\mathcal{Z}) - H(S_{i_t}|M,S_\mathcal{Z}) \text{(def. of conditional mutual information)}\\ -\leq H(S_{i_t}|S_\mathcal{Z}) \text{(entropy is non-negative)}\\ -\leq H(S_{i_t}|S_\mathcal{Z}) \text{(conditioning reduces entropy). \\ +&\leq H(S_{i_t}|S_\mathcal{Z}) \text{(entropy is non-negative)}\\ +&\leq H(S_{i_t}|S_\mathcal{Z}) \text{(conditioning reduces entropy)} \\ \end{aligned} $$ diff --git a/content/Math4201/Math4201_L35.md b/content/Math4201/Math4201_L35.md new file mode 100644 index 0000000..8cf210f --- /dev/null +++ b/content/Math4201/Math4201_L35.md @@ -0,0 +1,125 @@ +# Math4201 Topology I (Lecture 35) + +## Countability axioms + +### Kolmogorov classification + +Consider the topological space $X$. + +$X$ is $T_0$ means for every pair of points $x,y\in X$, $x\neq y$, there is one of $x$ and $y$ is in an open set $U$ containing $x$ but not $y$. + +$X$ is $T_1$ means for every pair of points $x,y\in X$, $x\neq y$, each of them have a open set $U$ and $V$ such that $x\in U$ and $y\in V$ and $x\notin V$ and $y\notin U$. (singleton sets are closed) + +$X$ is $T_2$ means for every pair of points $x,y\in X$, $x\neq y$, there exists disjoint open sets $U$ and $V$ such that $x\in U$ and $y\in V$. (Hausdorff) + +$X$ is $T_3$ means that $X$ is regular: for any $x\in X$ and any close set $A\subseteq X$ such that $x\notin A$, there are **disjoint open sets** $U,V$ such that $x\in U$ and $A\subseteq V$. + +$X$ is $T_4$ means that $X$ is normal: for any disjoint closed sets, $A,B\subseteq X$, there are **disjoint open sets** $U,V$ such that $A\subseteq U$ and $B\subseteq V$. + +
+Example + +Let $\mathbb{R}_{\ell}$ with lower limit topology. + +$\mathbb{R}_{\ell}$ is normal since for any disjoint closed sets, $A,B\subseteq \mathbb{R}_{\ell}$, $x\in A$ and $B$ is closed and doesn't contain $x$. Then there exists $\epsilon_x>0$ such that $[x,x+\epsilon_x)\subseteq A$ and does not intersect $B$. + +Therefore, there exists $\delta_y>0$ such that $[y,y+\delta_y)\subseteq B$ and does not intersect $A$. + +Let $U=\bigcup_{x\in A}[x,x+\epsilon_x)$ is open and contains $A$. + +$V=\bigcup_{y\in B}[y,y+\delta_y)$ is open and contains $B$. + +We show that $U$ and $V$ are disjoint. + +If $U\cap V\neq \emptyset$, then there exists $x\in A$ and $Y\in B$ such that $[x,x+\epsilon_x)\cap [y,y+\delta_y)\neq \emptyset$. + +This is a contradiction since $[x,x+\epsilon_x)\subseteq A$ and $[y,y+\delta_y)\subseteq B$. + +
+ +#### Theorem Every metric space is normal + +Use the similar proof above. + +
+Proof + +Let $A,B\subseteq X$ be closed. + +Since $B$ is closed, for any $x\in A$, there exists $\epsilon_x>0$ such that $B_{\epsilon_x}(x)\subseteq B$. + +Since $A$ is closed, for any $y\in B$, there exists $\delta_y>0$ such that $A_{\delta_y}(y)\subseteq A$. + +Let $U=\bigcup_{x\in A}B_{\epsilon_x/2}(x)$ and $V=\bigcup_{y\in B}B_{\delta_y/2}(y)$. + +We show that $U$ and $V$ are disjoint. + +If $U\cap V\neq \emptyset$, then there exists $x\in A$ and $Y\in B$ such that $B_{\epsilon_x/2}(x)\cap B_{\delta_y/2}(y)\neq \emptyset$. + +Consider $z\in B_{\epsilon_x/2}(x)\cap B_{\delta_y/2}(y)$. Then $d(x,z)<\epsilon_x/2$ and $d(y,z)<\delta_y/2$. Therefore $d(x,y)\leq d(x,z)+d(z,y)<\epsilon_x/2+\delta_y/2$. + +If $\delta_y<\epsilon_x$, then $d(x,y)<\delta_y/2+\delta_y/2=\delta_y$. Therefore $x\in B_{\delta_y}(y)\subseteq A$. This is a contradiction since $U\cap B=\emptyset$. + +If $\epsilon_x<\delta_y$, then $d(x,y)<\epsilon_x/2+\epsilon_x/2=\epsilon_x$. Therefore $y\in B_{\epsilon_x}(x)\subseteq B$. This is a contradiction since $V\cap A=\emptyset$. + +Therefore, $U$ and $V$ are disjoint. + +
+ +#### Lemma fo regular topological space + +$X$ is regular topological space if and only if for any $x\in X$ and any open neighborhood $U$ of $x$, there is open neighborhood $V$ of $x$ such that $\overline{V}\subseteq U$. + +#### Lemma of normal topological space + +$X$ is a normal topological space if and only if for any $A\subseteq X$ closed and any open neighborhood $U$ of $A$, there is open neighborhood $V$ of $A$ such that $\overline{V}\subseteq U$. + +
+Proof + +$\implies$ + +Let $A$ and $U$ are given as in the statement. + +So $A$ and $(X-U)$ are disjoint closed. + +Since $X$ is normal and $A\subseteq V\subseteq X$ and $V\cap W=\emptyset$. $X-U\subseteq W\subseteq X$. where $W$ is open in $X$. + +And $\overline{V}\subseteq (X-W)\subseteq U$. + +And $A\subseteq V$. + +The proof of reverse direction is similar. + +Let $A,B$ be disjoint and closed. + +Then $A\subseteq U\coloneqq X-B\subseteq X$ and $X-B$ is open in $X$. + +Apply the assumption to find $A\subseteq V\subseteq X$ and $V$ is open in $X$ and $\overline{V}\subseteq U\coloneqq X-B$. + +
+ +#### Proposition of regular and Hausdorff on subspaces + +1. If $X$ is a regular topological space, and $Y$ is a subspace. Then $Y$ with induced topology is regular. (same holds for Hausdorff) +2. If $\{X_\alpha\}$ is a collection of regular topological spaces, then their product with the product topology is regular. (same holds for Hausdorff) + +> [!CAUTION] +> +> The above does not hold for normal. + +Recall that $\mathbb{R}_{\ell}$ with lower limit topology is normal. But $\mathbb{R}_{\ell}\times \mathbb{R}_{\ell}$ with product topology is not normal. (In problem set 11) + +This shows that $\mathbb{R}_{\ell}$ is not metrizable. Otherwise $\mathbb{R}_{\ell}\times \mathbb{R}_{\ell}$ would be metrizable. Which could implies that $\mathbb{R}_{\ell}$ is normal. + +#### Theorem of metrizability + +If $X$ is normal and second countable, then $X$ is metrizable. + +> [!NOTE] +> +> - Every metrizable topological space is normal. +> - Every metrizable space is first countable. +> - But there are some metrizable space that is not second countable. +> +> Note that if $X$ is normal and first countable, then it is not necessarily metrizable. (Example $\mathbb{R}_{\ell}$) \ No newline at end of file diff --git a/content/Math4201/_meta.js b/content/Math4201/_meta.js index d874e5e..764d860 100644 --- a/content/Math4201/_meta.js +++ b/content/Math4201/_meta.js @@ -38,4 +38,5 @@ export default { Math4201_L32: "Topology I (Lecture 32)", Math4201_L33: "Topology I (Lecture 33)", Math4201_L34: "Topology I (Lecture 34)", + Math4201_L35: "Topology I (Lecture 35)", }