Fie
I
⊆
J
{\displaystyle \mathcal I \subseteq \mathcal J \!}
R
.
{\displaystyle \mathbb R. \!}
[1]
Lemă .
Fie
{
I
p
:
p
∈
N
}
⊆
I
.
{\displaystyle \{ I_p \; : \; p \in \mathbb N \} \subseteq \mathcal I. \!}
1) . Dacă
I
0
⊆
⋃
p
=
1
∞
I
p
,
{\displaystyle I_0 \subseteq \bigcup_{p=1}^{\infty} I_p, \!}
|
I
0
|
≤
∑
p
=
1
∞
|
I
p
|
.
{\displaystyle |I_0| \le \sum_{p=1}^{\infty} |I_p|. \!}
2) . Dacă
⋃
p
=
1
∞
I
p
⊆
I
0
{\displaystyle \bigcup_{p=1}^{\infty} I_p \subseteq I_0 \!}
I
p
∩
I
q
=
∅
,
∀
p
≠
q
,
{\displaystyle I_p \cap I_q = \varnothing, \; \forall p \neq q, \!}
∑
p
=
1
∞
|
I
p
|
≤
|
I
0
|
.
{\displaystyle \sum_{p=1}^{\infty} |I_p| \le |I_0|. \!}
Demonstraţie.
1). Dacă unul dintre intervalele
I
p
,
p
≥
1
{\displaystyle I_p, \; p \ge 1 \!}
|
I
p
|
=
+
∞
{\displaystyle |I_p| = + \infty \!}
a). Dacă primul interval
I
0
=
(
a
,
b
)
{\displaystyle I_0 = (a, b) \!}
ε
>
0
,
[
a
+
ε
,
b
−
ε
]
⊆
⋃
p
=
1
∞
(
a
p
,
b
p
)
{\displaystyle \varepsilon >0, \; [a+ \varepsilon, b-\varepsilon] \subseteq \bigcup_{p=1}^{\infty} (a_p, b_p) \!}
p
0
∈
N
∗
{\displaystyle p_0 \in \mathbb N^* \!}
[
a
+
ε
,
b
−
ε
]
⊆
⋃
k
=
1
p
0
(
a
k
,
b
k
)
.
{\displaystyle [a + \varepsilon, b - \varepsilon] \subseteq \bigcup_{k=1}^{p_0} (a_k, b_k). \!}
(*) (Dacă nu, pentru orice
p
∈
N
∗
{\displaystyle p \in \mathbb N^* \!}
x
p
∈
[
a
+
ε
,
b
−
ε
]
∖
⋃
k
=
1
p
(
a
k
,
b
k
)
.
{\displaystyle x_p \in [a + \varepsilon, b - \varepsilon] \setminus \bigcup_{k=1}^p (a_k, b_k). \!}
(
x
p
)
p
{\displaystyle (x_p)_p \!}
(
x
k
)
p
{\displaystyle (x_k)_p \!}
x
∈
[
a
+
ε
,
b
−
ε
]
.
{\displaystyle x \in [a + \varepsilon, b - \varepsilon]. \!}
p
1
∈
N
∗
{\displaystyle p_1 \in \mathbb N^* \!}
x
∈
I
p
1
;
{\displaystyle x \in I_{p_1}; \!}
I
p
1
{\displaystyle I_{p_1} \!}
vecinătate a lui x , există
p
2
∈
N
∗
,
p
2
>
p
1
{\displaystyle p_2 \in \mathbb N^*, \; p_2> p_1 \!}
x
k
p
∈
I
p
1
,
{\displaystyle x_{k_p} \in I_{p_1}, \!}
p
≥
p
2
.
{\displaystyle p \ge p_2. \!}
p
1
<
p
2
≤
k
p
2
,
{\displaystyle p_1 < p_2 \le k_{p_2}, \!}
x
k
p
2
∈
[
a
+
ε
,
b
−
ε
]
∖
⋃
k
=
1
p
(
a
k
,
b
k
)
.
{\displaystyle x_{k_{p_2}} \in [a + \varepsilon, b - \varepsilon] \setminus \bigcup_{k=1}^p (a_k, b_k). \!}
Relaţia (*) ne permite să reordonăm familia finită de intervale
{
(
a
k
,
b
k
)
:
k
=
1
,
2
,
⋯
,
p
0
}
{\displaystyle \{ (a_k, b_k) \; : \; k =1, 2, \cdots , p_0 \} \!}
a
1
<
a
+
ε
<
b
−
ε
<
b
p
0
.
{\displaystyle a_1 < a + \varepsilon < b -\varepsilon < b_{p_0}. \!}
b
−
a
−
2
ε
<
b
p
0
−
a
1
≤
∑
k
=
1
p
−
0
|
I
k
|
≤
∑
k
=
1
∞
|
I
k
|
.
{\displaystyle b-a - 2 \varepsilon < b_{p_0} - a_1 \le \sum_{k=1}^{p-0} |I_k| \le \sum_{k=1}^{\infty} |I_k|. \!}
ε
{\displaystyle \varepsilon\!}
b
−
a
=
|
I
0
|
≤
∑
p
=
1
∞
|
I
p
|
.
{\displaystyle b-a = |I_0| \le \sum_{p=1}^{\infty} |I_p|. \!}
I
0
=
(
a
,
+
∞
)
{\displaystyle I_0= (a, + \infty) \!}
n
∈
N
,
(
a
,
n
)
⊆
⋃
p
=
1
∞
I
p
.
{\displaystyle n \in \mathbb N, \; (a, n) \subseteq \bigcup_{p=1}^{\infty} I_p. \!}
n
−
a
≤
∑
k
=
1
∞
|
I
k
|
{\displaystyle n-a \le \sum_{k=1}^{\infty} |I_k| \! }
∑
k
=
1
∞
|
I
k
|
=
+
∞
=
|
I
0
|
.
{\displaystyle \sum_{k=1}^{\infty} |I_k| = + \infty = |I_0|. \!}
La fel se face raţionamentul şi în celelalte cazuri posibile pentru
I
0
.
{\displaystyle I_0. \!}
I
0
{\displaystyle I_0 \!}
|
I
0
|
=
+
∞
≥
∑
p
=
1
∞
|
I
p
|
.
{\displaystyle |I_0| = + \infty \ge \sum_{p=1}^{\infty} |I_p|. \!}
Să presupunem acum că
I
0
=
(
a
,
b
)
{\displaystyle I_0 = (a, b) \!}
I
p
=
(
a
p
,
b
p
)
,
p
∈
N
∗
,
{\displaystyle I_p= (a_p, b_p), \; p \in \mathbb N^*, \!}
n
∈
N
∗
,
⋃
p
=
1
n
I
p
⊆
I
0
.
{\displaystyle n \in \mathbb N^*, \; \; \bigcup_{p=1}^n I_p \subseteq I_0. \!}
I
1
,
I
2
,
⋯
,
I
n
{\displaystyle I_1, I_2, \cdots , I_n \!}
a
≤
a
1
≤
b
1
≤
a
2
≤
b
2
≤
⋯
≤
a
n
≤
b
n
≤
b
.
{\displaystyle a \le a_1 \le b_1 \le a_2 \le b_2 \le \cdots \le a_n \le b_n \le b. \!}
|
I
0
|
=
b
−
a
≥
(
b
1
−
a
1
)
+
(
b
2
−
a
2
)
+
⋯
+
(
b
n
−
a
n
)
=
∑
p
=
1
n
|
I
p
|
.
{\displaystyle |I_0| = b-a \ge (b_1-a_1)+ (b_2 - a_2) + \cdots + (b_n-a_n) = \sum_{p=1}^n |I_p|. \!}
Deoarece n este arbitrar în
N
∗
,
{\displaystyle \mathbb N^*, \!}
n
→
+
∞
{\displaystyle n \to + \infty \!}
Definiţie .
O mulţime
A
⊆
R
{\displaystyle A \subseteq \mathbb R \!}
deschisă dacă
A
=
∅
{\displaystyle A = \varnothing \!}
x
∈
A
,
{\displaystyle x \in A, \!}
I
∈
I
{\displaystyle I \in \mathcal I \!}
x
∈
I
⊆
A
.
{\displaystyle x \in I \subseteq A. \!}
τ
u
{\displaystyle \tau_u \!}
R
.
{\displaystyle \mathbb R. \!}
topologie pe
R
,
{\displaystyle \mathbb R, \!}
(
T
1
)
D
∩
G
∈
τ
u
,
∀
D
,
G
∈
τ
u
;
{\displaystyle \mathbf{(T_1)} \; \; D \cap G \in \tau_u, \; \forall D, G \in \tau_u; \!}
(
T
2
)
U
γ
∈
Γ
∈
τ
u
,
∀
{
D
γ
:
γ
∈
Γ
}
⊆
τ
u
;
{\displaystyle \mathbf{(T_2)} \; \; U_{\gamma \in \Gamma} \in \tau_u, \; \forall \{ D_{\gamma} \; : \; \gamma \in \Gamma \} \subseteq \tau_u; \!}
(
T
3
)
∅
,
R
∈
τ
u
.
{\displaystyle \mathbf{(T_3)} \; \; \varnothing, \mathbb R \in \tau_u. \!}
Vom spune că
τ
u
{\displaystyle \tau_u \!}
topologia uzuală pe
R
.
{\displaystyle \mathbb R. \!}
Note [ ]
↑ Prin
J
{\displaystyle \mathcal J \!}
R
.
{\displaystyle \mathbb R. \!}
Vezi şi [ ] Resurse [ ] 
![{\displaystyle \varepsilon >0,\;[a+\varepsilon ,b-\varepsilon ]\subseteq \bigcup _{p=1}^{\infty }(a_{p},b_{p})\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/035aa9b500c03e31939bd7a39d9bdbc38f9a8084)
![{\displaystyle [a+\varepsilon ,b-\varepsilon ]\subseteq \bigcup _{k=1}^{p_{0}}(a_{k},b_{k}).\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/fe18abdbb2875ed73f32b5d5feadf4abb0309191)
![{\displaystyle x_{p}\in [a+\varepsilon ,b-\varepsilon ]\setminus \bigcup _{k=1}^{p}(a_{k},b_{k}).\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a330fea03ab34745e9cb3dee03ee60c1aeb791f5)
![{\displaystyle x\in [a+\varepsilon ,b-\varepsilon ].\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/e650e8ffd38d68354f2acb41785676f0c9e6363c)
![{\displaystyle x_{k_{p_{2}}}\in [a+\varepsilon ,b-\varepsilon ]\setminus \bigcup _{k=1}^{p}(a_{k},b_{k}).\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/aafacdbfa62bfa572a93238cbc1da3e980be908f)
