Could you explain how to show 8.6 in detail? **Theorem 8.6.** Let $ C $ be a connected subset of the topological space $ X $. If $ D $ is a subset of $ X $ such that $ C \subseteq D \subseteq \overline{C} $, then $ D $ is connected.---**Definition.** Let $ X $ be a topological space. Then $ X $ is *connected* if and only if $ X $ is not the union of two disjoint non-empty open sets.**Definition.** Let $ X $ be a topological space. Subsets $ A, B $ in $ X $ are *separated* if and only if $ \overline{A} \cap B = A \cap \overline{B} = \emptyset $. Thus $ B $ does not contain any limit points of $ A $, and $ A $ does not contain any limit points of $ B $. The notation $ X = A \mid B $ means $ X = A \cup B $ and $ A $ and $ B $ are separated sets.---**Theorem 8.1.** The following are equivalent:1. $ X $ is connected.2. There is no continuous function $ f : X \rightarrow \mathbb{R}_{\text{std}} $ such that $ f(X) = \{0, 1\} $.3. $ X $ is not the union of two disjoint non-empty separated sets.4. $ X $ is not the union of two disjoint non-empty closed sets.5. The only subsets of $ X $ that are both closed and open in $ X $ are the empty set and $ X $ itself.6. For every pair of points $ p $ and $ q $ and every open cover $ \{U_\alpha\}_{\alpha \in \lambda} $ of $ X $ there exists a finite number of the $ U_\alpha $’s, $ \{U_{\alpha_1}, U_{\alpha_2}, U_{\alpha_3}, ..., U_{\alpha_n}\} $ such that $ p \in U_{\alpha_1} $, $ q \in U_{\alpha_n} $, and for each \( i < n \

Let C be a connected subset of the topological space X.Also given that D is a subset of X such that C ⊂ D ⊂ C—.Suppose that D = E ∪ F is a seperation of D.We know that if the sets E and F form a seperation of D and C is a connected subspace D, thenC lies entirely within either E or F.So, the set C must lie entirely in E or in F. Suppose that C ⊂ E then C—⊂ E—.Since E— and F are disjoint, D can not intersect F.This contradicts the facts that F is a non empty subset of D.Hence D is connected.

Answered: Theorem 8.6. Let C be a connected…

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Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that C CDCC, then D is connected.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Could you explain how to show 8.6 in detail?

**Theorem 8.6.** Let $ C $ be a connected subset of the topological space $ X $. If $ D $ is a subset of $ X $ such that $ C \subseteq D \subseteq \overline{C} $, then $ D $ is connected.

---

**Definition.** Let $ X $ be a topological space. Then $ X $ is *connected* if and only if $ X $ is not the union of two disjoint non-empty open sets.

**Definition.** Let $ X $ be a topological space. Subsets $ A, B $ in $ X $ are *separated* if and only if $ \overline{A} \cap B = A \cap \overline{B} = \emptyset $. Thus $ B $ does not contain any limit points of $ A $, and $ A $ does not contain any limit points of $ B $. The notation $ X = A \mid B $ means $ X = A \cup B $ and $ A $ and $ B $ are separated sets.

---

**Theorem 8.1.** The following are equivalent:

1. $ X $ is connected.

2. There is no continuous function $ f : X \rightarrow \mathbb{R}_{\text{std}} $ such that $ f(X) = \{0, 1\} $.

3. $ X $ is not the union of two disjoint non-empty separated sets.

4. $ X $ is not the union of two disjoint non-empty closed sets.

5. The only subsets of $ X $ that are both closed and open in $ X $ are the empty set and $ X $ itself.

6. For every pair of points $ p $ and $ q $ and every open cover $ \{U_\alpha\}_{\alpha \in \lambda} $ of $ X $ there exists a finite number of the $ U_\alpha $’s, $ \{U_{\alpha_1}, U_{\alpha_2}, U_{\alpha_3}, ..., U_{\alpha_n}\} $ such that $ p \in U_{\alpha_1} $, $ q \in U_{\alpha_n} $, and for each \( i < n \

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