We define connected components. We show how to use this as a tool to prove certain spaces are not homeomorphic.
00:00 Disclaimer
01:04 The main mantra of this lecture
1:58 - some motivating examples. [GL_n(R) & O_n(R)]
11:57 - R and R^n are not homeomorphic.
14:50 - Definition of connected components with examples.
20:10 - Connected Components are connected sets
22:02 - Each component is a maximal connected set
25:14 - Connected components are closed
26:13 - Summarize
27:15 - Rigorous examples in R and R^2 (with geometric intuition)
36:13 - Some topological lucid examples [IMPORTANT]
43:06 - Homeomorphism between alphabets.
48:40 - Outro
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