generated from smyalygames/quartz
26 lines
1.1 KiB
Markdown
26 lines
1.1 KiB
Markdown
# Definition
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A collection of subsets of $X$, called [[Open Sets|open sets]], such that:
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1. $X, \, \emptyset \in \tau$
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2. Any union of sets from $\tau$ will be in $\tau$.
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> [!info]-
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> $$y, z \in \tau \implies y \cup z \in \tau$$
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> $$x_{i} \in \tau \implies \cup_{i \in I} X_{i} \in \tau$$
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3. Any **finite** intersection of sets from $\tau$ will be in $\tau$.
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> [!info]-
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> $$y \cap z \in \tau$$
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> $$\cap_{i \in I} X_{i} \notin \tau$$
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# Examples
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> [!example]
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> The **topology induced by a metric** on $X$ is the collection of all unions of [[Ball|balls]].
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> [!example] Reasoning for having point/rule 3 in [[#Definition]]
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> Consider the topology on $\mathbb{R}$ induced by the usual distance.
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> $$B_{r}(x) = \langle x - r, x + r \rangle$$
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> Note:
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> $$\cap_{n \in \mathbb{N}} \langle -\frac{1}{n}, \frac{1}{n} \rangle = \{ 0\}$$
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> ($\cap_{n \in \mathbb{N}}$ is an infinite intersection of all numbers (in $\mathbb{N}$))
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> But the reason why this is not $= \{ 0, \varepsilon \}$ is a finite amount of intersections
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> $\varepsilon \notin \langle -\frac{1}{n}, \frac{1}{n} \rangle$ for $n \gt \frac{1}{\varepsilon}$
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