Difference between revisions of "Quotients (Topology)"
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| − | 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 $$a,b,c \in X$$: | 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 $$a,b,c \in X$$: | ||
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| − | 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. | # 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 $$f$$ is onto, then $$g$$ is onto. | ||
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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 $$∼$$. |
Revision as of 21:27, 21 May 2023
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.
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 $$∼$$.
