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You are here: Course Notes » Math 105: A second course to real analysis (2022 Spring) » Students Area » David Alcalay
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math105-s22:s:david_alcalay:start [2022/02/10 02:38]
davidtakesmath105 		math105-s22:s:david_alcalay:start [2026/02/21 14:41] (current)
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 Hello! My name is David Alcalay. I'm a third year pure-math major here at UC Berkeley. I grew up in Davis, California, and transferred a year ago from community college. Analysis has always been my favorite area of mathematics, so I'm really excited to be taking Math 105. I'm really interested in Cantor sets, and also the topic of constructible numbers (think ruler-and-compass).  Hello! My name is David Alcalay. I'm a third year pure-math major here at UC Berkeley. I grew up in Davis, California, and transferred a year ago from community college. Analysis has always been my favorite area of mathematics, so I'm really excited to be taking Math 105. I'm really interested in Cantor sets, and also the topic of constructible numbers (think ruler-and-compass). 
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-I have bipolar disorder, type 2, a mental illness characterized by periods of high energy, called hypomanic episodes, and periods of depression. I'm very passionate about the subject, and welcome any questions about my personal experiences with the illness, navigating UC Berkeley as a student with a mental illness, or more generally. 
  
 Outside of my education, I spend most of my time reading, hiking, doing art, and playing board games. Outside of my education, I spend most of my time reading, hiking, doing art, and playing board games.
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 My dream, upon finishing my education, is to become a research mathematician, specifically researching Cantor sets. My dream, upon finishing my education, is to become a research mathematician, specifically researching Cantor sets.
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 +==Homework==
 +  * {{ :math105-s22:s:david_alcalay:math_105_homework_1.pdf |Homework 1 (Incomplete)}}
 +  * {{ :math105-s22:s:david_alcalay:math_105_homework_2.pdf |Homework 2 (Rough Draft)}}
  
  
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 The following proof is from Measure and Category, second edition, by John C. Oxtoby. The following proof is from Measure and Category, second edition, by John C. Oxtoby.
  
-**Definition:** A subset $B \subseteq \mathbb{R}$ is called a Bernstein set if the following holds: For each subset $F \subseteq \mathbb{R}$ such that $F$ is an uncountable closed subset of $\mathbb{R}$, the intersection $B \cap F$ is nonempty.+**Definition:** A subset $B \subseteq \mathbb{R}$ is called a Bernstein set if the following holds: For each subset $F \subseteq \mathbb{R}$ such that $F$ is an uncountable closed subset of $\mathbb{R}$, the intersections $B \cap F$ and $B^c \cap F$ is nonempty.
  
-**Theorem (F. Bernstein):** There exists a set $B$ of real numbers such that $B$ and $B^c$ both meet every closed uncountable subset of the real numbers. (pg. 23)+**Theorem (F. Bernstein):** There exists a Bernstein set. (pg. 23)
  
 The proof relies on the Well-Ordering Principle, and uses transfinite induction to choose points from each uncountable closed subset of the reals. The proof relies on the Well-Ordering Principle, and uses transfinite induction to choose points from each uncountable closed subset of the reals.
math105-s22/s/david_alcalay/start.1644460702.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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