Quotients (Topology) - Revision history

Kfagerstrom at 21:34, 21 May 2023

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Kfagerstrom: Created page with "The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$..."

2023-05-21T21:18:41Z

Created page with "The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$..."

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The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the equivalence map.

The equivalence class of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

FBT = Function Building Theorem for quotient sets: Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The quotient topology, or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a quotient space of $$X$$, and the equivalence map $$q$$ is called the quotient map induced by $$∼$$.

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	The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.		The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.
		+
		+	[[Category:Point-Set Topology]]

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	# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.		# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

		+	<!-- Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). -->

−	(<!-- Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). -->) The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.	+	The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

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−	<!-- Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). --> The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.	+	(<!-- Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). -->) The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

@@ Line 1: / Line 1: @@
-The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes
+The '''quotient''' of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes
-The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the equivalence map.
+The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the '''equivalence map'''.
-The equivalence class of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.
+The '''equivalence class''' of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.
 An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
@@ Line 11: / Line 11: @@
-FBT = Function Building Theorem for quotient sets: Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
+'''FBT = Function Building Theorem for quotient sets:''' Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
 # There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
 # If $$f$$ is onto, then $$g$$ is onto.
@@ Line 17: / Line 17: @@
-Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The '''quotient topology''', or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map'' induced by $$∼$$.
+<!-- Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). --> The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

← Older revision		Revision as of 21:22, 21 May 2023
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−	Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The quotient topology, or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a quotient space of $$X$$, and the equivalence map $$q$$ is called the quotient map induced by $$∼$$.	+	Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The '''quotient topology''', or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ \| q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map'' induced by $$∼$$.