partial updates, continue on Riemannian manifold

content/Math401/Extending_thesis/Math401_R2.md

@@ -252,6 +252,32 @@ $$
 
 Not very edible for undergraduates.
 
+## Crash course on Riemannian Geometry
+
+> This section is designed for stupids like me skipping too much essential materials in the book.
+
+### Manifold
+
+Unexpectedly, a good definition of the manifold is defined in the topology I.
+
+Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2.
+
+#### Definition of m-manifold
+
+An $m$-manifold is a Hausdorff space $X$ with a countable basis such that each point of $x$ of $X$ has a neighborhood <text style="color: red;"> homeomorphic</text> to an open subset of $\mathbb{R}^m$.
+
+Example is trivial that 1-manifold is a curve and 2-manifold is a surface.
+
+#### Theorem of imbedded space
+
+If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
+
+This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
+
+### Riemannian manifold
+
+
+
 ## Crash course on Riemannian manifolds
 
 > This part might be extended to a separate note, let's check how far we can go from this part.

content/Math401/Extending_thesis/Math401_S2.md

@@ -1,6 +1,6 @@
 # Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states.
 
-## Majorana representation of quantum states
+## Majorana stellar representation of quantum states
 
 > [!TIP]
 >
@@ -26,6 +26,6 @@ Using stereographic projection of each root we can get a unordered collection of
 
 > [!NOTE]
 >
-> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana representation of quantum states.
+> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana stellar representation of quantum states.
 >
 > Read Chapter 5 and 6 of [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) for more details.

content/Math4201/Math4201_L8.md

@@ -15,7 +15,7 @@ Let $X=\mathbb{R}$ with standard topology.
 
 Let $A=(0,1)$, then set of limit points of $A$ is $[0,1]$.
 
-Let $A=\left{\frac{1}{n}\right}_{n\in \mathbb{N}}$, then set of limit points of $A$ is $\{0\}$.
+Let $A=\left\{\frac{1}{n}\right\}_{n\in \mathbb{N}}$, then set of limit points of $A$ is $\{0\}$.
 
 Let $A=\{0\}\cup (1,2)$, then set of limit points of $A$ is $[1,2]$