Notes in Topology

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Status	Last Update	Fields
Published	04/24/2024	If we have [object] with [properties], it is [definition] if:
Published	04/24/2024	Given (so and so) we may do (such and such) as follows:
Published	04/24/2024	(property 1) is the same as
Published	04/24/2024	[Sample Generic (Reverse)]
Published	04/24/2024	idk I've never used this one but it might be useful
Published	04/24/2024	Given [conditions]
Published	04/24/2024	$f:X\to Y$ is a function$B \subseteq Y$
Published	01/24/2024	test
Published	04/24/2024	$X$ is a set$ \mathcal T \in \mathcal P (X)$ such that
Published	02/11/2024	Two sets, $A$ and $B$, have the same $\textbf{cardinality}$ if and only if
Published	04/24/2024	$A$ and $B$ are sets
Published	04/24/2024	$(X ,\mathcal T)$ is a topological space$p \in X$
Published	04/24/2024	$X $ is a topological space
Published	04/24/2024	$Y \hookrightarrow X$ are topological spaces$C = \set{S_i}_{i\in I}$ is a collection of subsets of $X$
Published	04/24/2024	$Y \hookrightarrow X$ are topological spaces$C = \set{S_i}_{i\in I}$ is a collection of open subsets of $X$
Published	04/24/2024	$Y \hookrightarrow X$ are topological spaces$C$ is an open cover of $Y$
Published	02/11/2024	$X$ is a finite set.
Published	04/24/2024	$X$ is a set.
Published	04/24/2024	$X$ is a set.
Published	04/24/2024	$X$ is a set.
Published	04/24/2024	Standard definition:$[\forall y \in y][\exists x \in X][f(x) = y]$
Published	04/24/2024	Standard definition; $f$ is left-unique:$[\forall x_1,x_2 \in X][f(x_1)=f(x_2) \implies x_1=x_2]$
Published	04/24/2024	Let $f:X\to Y$ be a function and$A\subseteq X$
Published	04/24/2024	$A $ and $B $ are sets
Published	04/24/2024	$A $ and $B $ are sets
Published	04/24/2024	Let $I $ be an indexing set Let $\langle S_i \rangle_{i \in I} $ be a family of sets indexed by $I$
Published	04/24/2024	Let $I $ be an indexing set Let $\langle S_i \rangle_{i \in I} $ be a family of sets indexed by $I$
Published	04/24/2024	$f:X\to Y$ is a function
Published	04/24/2024	Let $\sim $ be a homogenous binary relation over a set $A$
Published	04/24/2024	Let $\sim $ be an equivalence relation on a set $S$
Published	04/24/2024	Standard Definition: $S \in \mathcal T$
Published	04/24/2024	The smallest closed set containing $A$:$\overline A = \bigcap_{A \subseteq S \subseteq C} S$
Published	04/24/2024	The collection of subsets $U$ of $\mathbb{R}^n$ fulfilling the property that, for all $p\in U$, there exists and open b…
Published	04/24/2024	$X$ is an arbitrary set
Published	04/24/2024	$X$ is a set
Published	04/24/2024	$X$ is a set
Published	04/24/2024	$\{X_i\}_{i \in \mathbb N}$ is a countable family of countable sets
Published	04/24/2024	For any set $X$, $\|X\| < \|\mathcal P (X)\|$
Published	04/24/2024	$A$ and $B$ are sets
Published	04/24/2024	$A$ and $B$ are sets
Published	04/24/2024	$X$ is a set
Published	04/24/2024	A point $p\in X$ such that every open neighborhood $U $ of $p$ contains some point of $A$ distinc…
Published	04/24/2024	A point $p \in A$ such that $p\in A$ but $p$ is not a limit point of $A$.
Published	04/24/2024	$A$ is equal to its own closure: $\overline A = A$
Published	04/24/2024	$(X , \mathcal T)$ is a topological space$S \subset \mathcal T$
Published	04/24/2024	$S$ is a basis for some topology on $X$
Published	04/24/2024	The topology on $\mathbb R $ consisting of all sets of the form $[a,b)$
Published	04/24/2024	$T,T'$ are topologies on the same underlying set.
Published	04/24/2024	$T,T'$ are topologies on the same underlying set.
Published	04/24/2024	The topology $\mathcal T$ on the set $\mathbb{R}\cup \{0',0''\}$ such that:
Published	04/24/2024	The topology $\mathcal T$ on the upper half plane $\{(x,y) \in \mathbb{R}^n\| x,y \in \mathbb{R}, y \ge 0\}$ such that:
Published	04/24/2024	The topology on $\mathbb Z $ whose basis elements are arithmetic progressions
Published	04/24/2024	$S$ is a basis for $\mathcal T$
Published	04/24/2024	$(X , \mathcal T)$ is a topological space$S \subset \mathcal P (X)$
Published	04/24/2024	$(X , \mathcal T)$ is a topological space$S \subset X$
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space.
Published	04/24/2024	$(X, \mathcal T) $ is a topological space$P$ is a topological property
Published	04/24/2024	$(X, \mathcal T) $ is a topological space$A \subset X$
Published	04/24/2024	$(X, \mathcal T) $ is a topological space
Published	04/24/2024	$(X, \mathcal T) $ is a topological space
Published	04/24/2024	$(X, \mathcal T) $ is a topological space$p \in X$A collection of open sets $\{U_\alpha\}_{\alpha \in \lambda}$ such that,
Published	04/24/2024	$(X, \mathcal T) $ is a topological space
Published	04/24/2024	$(X, \mathcal T) $ is a topological space$A \subset \mathcal P(X)$ is a collection of subsets of $X$
Published	04/24/2024	$(X, \mathcal T) $ is a topological space
Published	04/24/2024	$(X, \mathcal T) $ is a topological space${\cal B} = \{ B_\alpha\}_{\alpha\in\lambda}$
Published	04/24/2024	$(X, \mathcal T) $ is a topological space $\mathcal B = \{ B_\alpha\}_{\alpha\in\lambda}$ is a cover of $X$\({\cal C} = \{ C…
Published	04/24/2024	The preimage of every open set in $Y$ is open in $X$
Published	04/24/2024	$X,Y $ are topological spaces
Published	04/24/2024	$X,Y $ are topological spaces
Published	04/24/2024	$f$ is bijective and $f^{-1} $ is continuous
Status	Last Update	Fields