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MATH 4800-001 FALL 2024 - Week 2 - Metric Spaces 2
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This is MATH 4800-001, the introduction to mathematics research for undergraduates class at the University of Utah, which focuses on fractal geometry and dynamics.
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Table of Contents::
00:00:00 1 of 2
00:00:22 on weekly reports 1-14
00:09:03 uses of Holder condition: absolute continuity of stable manifolds, dimension preservation
00:11:45 homeomorphism, bi-uniform, bi-Lipschitz, bi-Holder homeomorphism
00:12:55 topological, uniform, Lipschitz, Holder equivalence of metrics
00:13:47 examples: metrics are continuous relative to themselves
00:22:38 distance between a subset and a point
00:26:41 exercise
00:27:18 exercise
00:28:33 exercise
00:29:06 exercise: topology of the extended nonnegative real numbers [0,oo]
00:30:30 level sets of metrics: open balls, closed balls, spheres etc.
00:34:55 exercise: balls relative to the discrete metric
00:38:13 level sets of metrics continued: subsets at the center
00:40:34 example: Vakil's doodles
00:42:22 relation to attractors of IFSs
00:43:29 sequences
00:44:49 subsequences: sequences obtained by sampling other sequences
00:46:57 example
00:50:03 convergent sequences
00:53:36 notation for a sequence converging, limit
00:54:01 some logic: converting statements into sets; Borel measurable subsets
00:59:44 exercise: continuity via convergent sequences
01:01:44 Cauchy sequences
01:05:27 proposition: any convergent sequence is Cauchy
01:06:11 proof of prop
01:10:08 epsilon logistics
01:11:59 complete metric spaces: sequence is Cauchy iff convergent
01:14:18 real numbers is complete, rational numbers is not
01:16:20 2 of 2
01:17:02 various categories ( = contexts): smooth, rough smooth, uniform, topological, measurable, amorphous
01:21:42 where metrizability fits in
01:23:29 exercise: uniformly continuity and Cauchy sequences
01:25:13 exercise
01:25:54 exercise
01:26:55 fundamental notions of topology: open subsets, closed subsets
01:30:01 examples: open intervals are open, closed intervals are closed
01:33:32 examples: some subsets are neither open nor closed, some subsets are both open and closed
01:37:04 exercise
01:39:24 exercise: continuity via open subsets
01:40:48 preimage of a set under a function
01:41:49 exercise: closedness via sequences
01:43:08 example
01:45:34 interior and closure operations
01:47:50 example
01:48:53 what largest and smallest mean for sets
01:51:29 exercise
01:52:58 example
01:53:39 dense subsets
01:54:26 example: rationals is dense in reals
01:55:04 compact subsets
01:59:33 exercise [when X=Rn]
02:00:30 bounded subsets, totally bounded subsets
02:04:18 exercise
02:05:38 exercise
02:09:03 exercise
02:10:03 factorization of concepts
02:10:44 theorem: in a complete metric space, a subset is compact iff it is closed and totally bounded
02:13:07 corollary: in Rn, a subset is compact iff it is closed and bounded
02:13:58 proof of theorem, only if side
---
Links:
Vakil's Doodles:
---
License:
CC BY-NC-SA 4.0
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License
Alp Uzman
This is MATH 4800-001, the introduction to mathematics research for undergraduates class at the University of Utah, which focuses on fractal geometry and dynamics.
---
Table of Contents::
00:00:00 1 of 2
00:00:22 on weekly reports 1-14
00:09:03 uses of Holder condition: absolute continuity of stable manifolds, dimension preservation
00:11:45 homeomorphism, bi-uniform, bi-Lipschitz, bi-Holder homeomorphism
00:12:55 topological, uniform, Lipschitz, Holder equivalence of metrics
00:13:47 examples: metrics are continuous relative to themselves
00:22:38 distance between a subset and a point
00:26:41 exercise
00:27:18 exercise
00:28:33 exercise
00:29:06 exercise: topology of the extended nonnegative real numbers [0,oo]
00:30:30 level sets of metrics: open balls, closed balls, spheres etc.
00:34:55 exercise: balls relative to the discrete metric
00:38:13 level sets of metrics continued: subsets at the center
00:40:34 example: Vakil's doodles
00:42:22 relation to attractors of IFSs
00:43:29 sequences
00:44:49 subsequences: sequences obtained by sampling other sequences
00:46:57 example
00:50:03 convergent sequences
00:53:36 notation for a sequence converging, limit
00:54:01 some logic: converting statements into sets; Borel measurable subsets
00:59:44 exercise: continuity via convergent sequences
01:01:44 Cauchy sequences
01:05:27 proposition: any convergent sequence is Cauchy
01:06:11 proof of prop
01:10:08 epsilon logistics
01:11:59 complete metric spaces: sequence is Cauchy iff convergent
01:14:18 real numbers is complete, rational numbers is not
01:16:20 2 of 2
01:17:02 various categories ( = contexts): smooth, rough smooth, uniform, topological, measurable, amorphous
01:21:42 where metrizability fits in
01:23:29 exercise: uniformly continuity and Cauchy sequences
01:25:13 exercise
01:25:54 exercise
01:26:55 fundamental notions of topology: open subsets, closed subsets
01:30:01 examples: open intervals are open, closed intervals are closed
01:33:32 examples: some subsets are neither open nor closed, some subsets are both open and closed
01:37:04 exercise
01:39:24 exercise: continuity via open subsets
01:40:48 preimage of a set under a function
01:41:49 exercise: closedness via sequences
01:43:08 example
01:45:34 interior and closure operations
01:47:50 example
01:48:53 what largest and smallest mean for sets
01:51:29 exercise
01:52:58 example
01:53:39 dense subsets
01:54:26 example: rationals is dense in reals
01:55:04 compact subsets
01:59:33 exercise [when X=Rn]
02:00:30 bounded subsets, totally bounded subsets
02:04:18 exercise
02:05:38 exercise
02:09:03 exercise
02:10:03 factorization of concepts
02:10:44 theorem: in a complete metric space, a subset is compact iff it is closed and totally bounded
02:13:07 corollary: in Rn, a subset is compact iff it is closed and bounded
02:13:58 proof of theorem, only if side
---
Links:
Vakil's Doodles:
---
License:
CC BY-NC-SA 4.0
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License
Alp Uzman
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