Differences

This shows you the differences between two versions of the page.

--- math104-s21:s:ryotainagaki [2021/05/12 08:44]
73.15.53.135
+++ math104-s21:s:ryotainagaki [2026/02/21 14:41] (current)
@@ Line 8: / Line 8: @@
 . Sets and sequences. (Ch 1- 10 in Ross) \\
 . Subsequences, Limsup, Liminf. (Ch 10 - 12 in Ross) \\
-. Compactness and Topology 101 (Ch 13 in Ross, Chapter 2 in Rudin) \\
+. Topology (Ch 13 in Ross, Chapter 2 in Rudin) \\
 . Series (Ch 14) \\
 . Continuity, Uniform Continuity, Uniform Convergence (Chapter 4, 7 in Rudin)\\
@@ Line 37: / Line 37: @@
 More mathematical way of putting this (supremum is sup and infimum is inf):
-$$\sup S = x \iff (\forall s \in S, s \leq x) \land (\forall y < x, \exists s_b \in S: y < s_b < x)$$
+$$\sup S = x \iff (\forall s \in S, s \leq x) \land (\forall y < x, \exists s_b \in S: y < s_b \leq x)$$
-$$\inf S = x \iff (\forall s \in S, s \geq x) \land (\forall y > x, \exists s_b \in S: y > s_b > x)$$
+$$\inf S = x \iff (\forall s \in S, s \geq x) \land (\forall y > x, \exists s_b \in S: y > s_b \geq x)$$
 ** E.g.**
- $$\sup \{x * y: x, y \in \{ 1, 2, 3, 4\}\} = 16$$ since $\{x * y: x, y \in \{ 1, 2, 3, 4\}\} = \{1, 2, 3, 4, 6, 8, 9, 12, 16\}$. All elements in the set are less than or equal to 16 and for any real number x less than 16, there exists a number in the set greater than x and less than or equal to 16; hence making 16 is an upper bound. Consider that for any z under 16 we always have an element in the set that is greater than z, e.g. 16. This makes 16 a ** minimum upper bound ** of the set.
+ $$\sup \{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = 16$$ since $\{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = \{1, 2, 3, 4, 6, 8, 9, 12, 16\}$. All elements in the set are less than or equal to 16 and for any real number x less than 16, there exists a number in the set greater than x and less than or equal to 16; hence making 16 is an upper bound. Consider that for any z under 16 we always have an element in the set that is greater than z, e.g. 16. This makes 16 a ** minimum upper bound ** of the set.
-$$\inf \{x * y: x, y \in \{ 1, 2, 3, 4\}\} = 1$$. All elements in the set are greater than or equal to 1; hence making it an upper bound. Consider that for any z over 1 we always have an element in the set that is less than z, e.g. 1. This makes 1 a ** maximum lower bound ** of the set.
+$$\inf \{x \cdot y: x, y \in \{ 1, 2, 3, 4\}\} = 1$$. All elements in the set are greater than or equal to 1; hence making it an upper bound. Consider that for any z over 1 we always have an element in the set that is less than z, e.g. 1. This makes 1 a ** maximum lower bound ** of the set.
 ** E.g. **
@@ Line 184: / Line 184: @@
 Therefore $E' = \emptyset$. And therefore $E = \overline{E}$ and thus $[0, 0.5]$ is closed. This illustrates that just because $E$ is open does not necessarily mean that it is not closed.
-==== Other Definitions ====
+==== Useful results on Open Sets Closed Sets ====
+  - A set is open iff its complement is closed.
+  - Given that $\bigcup S_i$ is a union of open sets, $\bigcup S_i$ is open.
+  - Given that $\bigcap S_i$ is the intersection of closed sets, $\bigcap S_i$ is closed.
+  - Given that $\bigcup S_i$ is a union of **finitely many** closed sets, $\bigcup S_i$ is closed.
+  - Given that $\bigcap S_i$ is the intersection of open sets, $\bigcap S_i$ is open.
+==== Induced Topology ====
+Sometimes, we want to create our own metric space and transfer over properties. How these properties transfer over can be summarized by the following drawing (credits to Peng Zhou).
+{{ math104-s21:s:inducedtopology.jpg?400 |}}
+In this diagram there are two ways to create induced topology. We can either obtain the topology from an induced metric or from the old set itself.
+One key idea to note is that (and I quote from course notes) **Given $A \subseteq S$ and (S, d) is a metric space, we can equip A with an induced metric. $E \subseteq A$ is open iff $\exists$ open set $ E_2 \subseteq S$ such that $E = E_2 \cap S$. From how a complement of an open set in ambient space is closed, we can use this theorem to tell alot topologically about the set.**
+==== Other Basic Topology Definitions ====
   - A Set is ** countable ** iff there exists a one to one mapping between the elements of the set to the set of natural numbers.
   - An ** open ball ** (or neighborhood as referred to in Rudin), about point x and with radius r is $B(r; x) = \{y \in M: d(y, x) < r\}$ where (M, d) is the ambient metric space of x.
@@ Line 190: / Line 209: @@
   - $S$ is bounded iff $\exists r: \forall p, q \in S, d(p, q) \leq r$.
   - $S$ is dense in $X$ iff every element in X is in $S$ or is a limit point of $S$.
+  - $S$ is a perfect set iff $S$ is closed and $S$ does not have any isolated points.
   - $diam(S)$, the diameter of $S$ is given to be $diam(S) = \sup_{x, y \in S} d(x, y)$.
 ** E.g.: ** We can say that in the set $\{\frac{1}{n}: n in \mathbb{N}\}$ has a limit point $0$ since we know that $\lim \frac{1}{n} = 0$ and therefore $\forall \epsilon > 0, \exists N \in \mathbb{N}: \forall n \geq N, d(\frac{1}{n}, 0) = |\frac{1}{n} - 0| < \epsilon$. Therefore, $\forall \epsilon > 0, B(\epsilon; x)$
-=== Useful results on Open Sets Closed Sets ====
-  - A set is open iff its complement is closed.
-  - Given that $\bigcup S_i$ is a union of open sets, $\bigcup S_i$ is open.
-  - Given that $\bigcap S_i$ is the intersection of closed sets, $\bigcap S_i$ is closed.
-  - Given that $\bigcup S_i$ is a union of **finitely many** closed sets, $\bigcup S_i$ is closed.
-  - Given that $\bigcap S_i$ is the intersection of open sets, $\bigcap S_i$ is open.
@@ Line 401: / Line 415: @@
 Given real functions $f, g$ continuous over interval $[x_1, x_2]$ and differentiable over $(x_1, x_2)$. We know that $\exists x \in (x_1, x_2): (f(x_2) - f(x_1))g'(x) = (g(x_2) - g(x_1))(f'(x))$.
+=== Intermediate Value Theorems for Derivatives ===
+In order to make an analog to the intermediate value theorem for continuous functions, we use make the following statement:
+** Given that f is differentiable over $[a, b]$, $f'(a) < f'(b)$, and $\mu$ is in $(f'(a), f'(b))$, I know that $\exists c \in (a, b): f'(x_0) = \mu$.**
+One important thing to note here is that this ** does not result from the intermediate value theorem for continuous functions. ** Rather, it originates from the idea of global minima. The proof is very simple:
+Consider $g(x) = f(x) - \mu x$. We know $g'(a) = f'(a) - \mu < 0$ and $g'(b) = f'(a) - \mu > 0$. Now,since $[a, b]$ is compact, we know that there exists a global minimum for $f$. It cannot be $a, b$ since $f'(a) < 0, f'(b) > 0$ are nonzero. Therefore, we must say that the global minimum for $f$ must be $x_0 \in (a, b)$. Since $f$ is continuous on $[a, b]$ it must be the case that $x_0$ is a local minimum as well. Thus, $f'(x_0) = 0$.
 ==== L'Hopital's Rule ====

Lecture Notes

User Tools

Site Tools

Differences

Page Tools