generated from smyalygames/quartz
32 lines
1.0 KiB
Markdown
32 lines
1.0 KiB
Markdown
---
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lecture: 6
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date: 2025-01-27
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---
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# Open Sets
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The concept of [[Ball|balls]] is required to understand what an [[Open Sets|open sets]] are.
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## Definition
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The definition can be found in [[Open Sets]].
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In other words ( "):" ), for any open $\overbrace{A}^{\in x} \exists N$ such that $x_{n} \in A \; \forall n \gt N$.
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In fact, [[Open Sets|open sets]] are unions of [[Ball|balls]].
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# Topological Space
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A [[Topological Space]] $X$ is a set with a [[Topology]] $\tau$.
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> [!example] Trivial [[Topology]]
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> $X$ is the set. $\tau = \{ \emptyset, X \}$.
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>
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> Therefore $x \neq y$, then the only neighbour is $X$
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> $X \cap X = X$.
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> [!example] Discrete [[Topology]]
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> $X$ is the set. $\tau = \wp(X)$.
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>
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> $\{ x \} \in \tau$
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> $X \, \text{compact} \iff X \, \text{finite}$
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>
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> This topological space is always [[Hausdorff]] as it includes all the points on $X$ are included
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> > [!note]
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> > $$A = \{ x \}, \; B = \{ y \} \; A \cup B = \emptyset$$
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>
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> $$d(x, y) = \begin{cases} 1, & \text{if}\ x \neq y\\ 0, & \text{if} \, x = y\end{cases}$$
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