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Title: Hausdorff Spaces
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Series: Manifolds
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YouTube-Title: Manifolds 3 | Hausdorff Spaces
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Bright video: https://youtu.be/Ty3ZLl7cOI0
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Dark video: https://youtu.be/LEeRgxRsqSc
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mf03_sub_eng.srt missing
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Other languages: German version
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $(X, \mathcal{T}) $ be a topological space. What is the correct definition for the convergence for a sequence $(a_n)_{n \in \mathbb{N}}$ in $X$?
A1: $ (\exists a \in X) $ $~ (\forall U \in \mathcal{T} \text{ with } a \in U)$ $~ (\exists N \in \mathbb{N}) ~ (\forall n \geq N) ~:$ $$a_n \in U $$
A2: $ (\exists a \in X) $ $~ (\forall U \in \mathcal{T} \text{ with } a \in U) $ $~ (\forall N \in \mathbb{N}) $ $~ (\exists n \geq N) ~:$ $$a_n \in U $$
A3: $ (\exists a \in X) $ $~ (\exists U \in \mathcal{T} \text{ with } a \in U) $ $~ (\exists N \in \mathbb{N}) $ $~ (\forall n \geq N) ~:$ $$a_n \in U $$
A4: $ (\forall a \in X) $ $~ (\forall U \in \mathcal{T} \text{ with } a \in U) $ $~ (\exists N \in \mathbb{N}) $ $~ (\forall n \geq N) ~:$ $$a_n \in U $$
Q2: Let $(X, \mathcal{T}) $ be a topological space. What do we need such that we call this space a Hausdorff space?
A1: For two different points $x,y \in X$, we find sets $U_x, U_y \subseteq X$ with $U_x \cap U_y = \emptyset$.
A2: For two different points $x,y \in X$, we find sets $U_x, U_y \in \mathcal{T}$ with $U_x \cap U_y = \emptyset$.
A3: For two different points $x,y \in X$, we find sets $U_x, U_y \subseteq X$ with $U_x \cap U_y \neq \emptyset$.
Q3: Let $(X, \mathcal{T}) $ be a topological space, where the topology $\mathcal{T}$ is induced by a metric $d$ on $X$. This means that $U \in \mathcal{T}$ if and only if $U$ is open in the metric space $(X,d)$. Is $(X, \mathcal{T}) $ a Hausdorff space?
A1: Yes, always!
A2: No, never!
A3: There are positive examples and counterexamples.
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Last update: 2024-10
