Topological space

$Topological space$

(X, U_{X})

is topological space:

U_{X}

is a topology of

X

A

is a subspace topology of

X

O_{A} := {O \cap A ∣ O \in O}

(X, O)

の部分空間:

(A, O_{A})

(X, O_{X})

is disconnected:

\exists O_{1}, O_{2} \in O_{X} (O_{1}, O_{2} \neq = ϕ, O_{1} \cap O_{2} = ϕ, O_{1} \cup O_{2} = X)

(X, O_{X})

is connected space:

(X, O_{X})

is not disconnected

(A, O_{A})

が

(X, O_{X})

is disconnected subspace:

(A, O_{A})

is Disconnected subspace

(A, O_{A})

が

(X, O)

is a disconnected subspace

⟺

\exists O_{1}, O_{2} \in O_{A} (O_{1} \cap A \neq = ϕ, O_{2} \cap A \neq = ϕ, O_{1} \cap O_{2} \cap A = ϕ, A \subset O_{1} \cap O_{2})

(A, O_{A})

is connected subspace of

(X, O_{X})

:

(A, O_{A})

is not disconnected space

f

is continuous

\Rightarrow (f (A), O_{f (A)})

is connected subspace of

(Y, O_{Y})

Base of

X_{1} \times X_{2}

:

O_{1} \times O_{2}

Product topology of

X_{1} \times X_{2}

O_{1} ♮ O_{2}

:=

{λ \in Λ ⋃ O_{λ} \times O_{λ}^{'} ∣ O_{λ} \times O_{λ}^{'} \in O_{1} \times O_{2}}

(X_{1}, O_{1})

と

(X_{2}, O_{2})

product space:

(X_{1} \times X_{2}, O_{1} ♮ O_{2})

(X, O_{X})

is Quotient topology by

f

O^{f} := {O \subset Y ∣ f^{- 1} (O) \in O_{X}}

(X, O_{X})

is a quotient space by

f

:

(Y, O^{f})

Natural projection to

X / R

π : X \to X / R, x \mapsto [x]