Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained from RK by collapsing the set K to a point. Prove that Y satisfies the T1 axiom, but is not Hausdorff.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Text:**

Recall that \( \mathbb{R}_K \) denotes the real line with the \( K \)-topology. Let \( Y \) denote the quotient space obtained from \( \mathbb{R}_K \) by collapsing the set \( K \) to a point. Prove that \( Y \) satisfies the \( T_1 \) axiom, but is not Hausdorff.

**Explanation:**

This text discusses a concept from topology, a branch of mathematics. It introduces:

- **\( \mathbb{R}_K \)**: The real number line equipped with a topology defined by a set \( K \).
- **Quotient Space \( Y \)**: A topological space derived by collapsing a subset \( K \) from \( \mathbb{R}_K \) into a single point.
- **\( T_1 \) Axiom**: A property of a topological space where for any two distinct points, each has a neighborhood not containing the other.
- **Hausdorff Condition**: A stronger separation property where any two distinct points have disjoint neighborhoods.

The task is to demonstrate that the quotient space \( Y \) adheres to the \( T_1 \) condition but does not fulfill the Hausdorff separation criterion.
Transcribed Image Text:**Text:** Recall that \( \mathbb{R}_K \) denotes the real line with the \( K \)-topology. Let \( Y \) denote the quotient space obtained from \( \mathbb{R}_K \) by collapsing the set \( K \) to a point. Prove that \( Y \) satisfies the \( T_1 \) axiom, but is not Hausdorff. **Explanation:** This text discusses a concept from topology, a branch of mathematics. It introduces: - **\( \mathbb{R}_K \)**: The real number line equipped with a topology defined by a set \( K \). - **Quotient Space \( Y \)**: A topological space derived by collapsing a subset \( K \) from \( \mathbb{R}_K \) into a single point. - **\( T_1 \) Axiom**: A property of a topological space where for any two distinct points, each has a neighborhood not containing the other. - **Hausdorff Condition**: A stronger separation property where any two distinct points have disjoint neighborhoods. The task is to demonstrate that the quotient space \( Y \) adheres to the \( T_1 \) condition but does not fulfill the Hausdorff separation criterion.
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