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math104-s21:s:ryotainagaki [2021/05/12 20:08]
73.15.53.135 		math104-s21:s:ryotainagaki [2026/02/21 14:41] (current)
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 1. Sets and sequences. (Ch 1- 10 in Ross) \\ 1. Sets and sequences. (Ch 1- 10 in Ross) \\
 2. Subsequences, Limsup, Liminf. (Ch 10 - 12 in Ross) \\ 2. Subsequences, Limsup, Liminf. (Ch 10 - 12 in Ross) \\
-3. Compactness and Topology 101 (Ch 13 in Ross, Chapter 2 in Rudin) \\+3. Topology (Ch 13 in Ross, Chapter 2 in Rudin) \\
 4. Series (Ch 14) \\ 4. Series (Ch 14) \\
 5. Continuity, Uniform Continuity, Uniform Convergence (Chapter 4, 7 in Rudin)\\ 5. Continuity, Uniform Continuity, Uniform Convergence (Chapter 4, 7 in Rudin)\\
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 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. 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.   - 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.   - 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.
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   - $S$ is bounded iff $\exists r: \forall p, q \in S, d(p, q) \leq r$.   - $S$ 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 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 $diam(S) = \sup_{x, y \in S} d(x, y)$.   - $diam(S)$, 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, $\forall \epsilon > 0, B(\epsilon; x)$ ** 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. 
  
  
math104-s21/s/ryotainagaki.1620850131.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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