Every Hausdorff space is hereditarily Hausdorff.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you explain how to show 4.19 in easiest possible way(in very detail)?

**Theorem 4.19.** Every Hausdorff space is hereditarily Hausdorff.

*X* is **Hausdorff**, or a **T₂-space**, if and only if for every pair *x, y* of distinct points, there are **disjoint** open sets *U, V* such that *x* ∈ *U* and *y* ∈ *V*.

**Definition.** Let *P* be a topological property (such as *T₁*, Hausdorff, etc.). A topological space *X* is **hereditarily P** if and only if for each subspace *Y* of *X*, the space *Y* has property *P* when *Y* is given the relative topology from *X*.
Transcribed Image Text:**Theorem 4.19.** Every Hausdorff space is hereditarily Hausdorff. *X* is **Hausdorff**, or a **T₂-space**, if and only if for every pair *x, y* of distinct points, there are **disjoint** open sets *U, V* such that *x* ∈ *U* and *y* ∈ *V*. **Definition.** Let *P* be a topological property (such as *T₁*, Hausdorff, etc.). A topological space *X* is **hereditarily P** if and only if for each subspace *Y* of *X*, the space *Y* has property *P* when *Y* is given the relative topology from *X*.
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