Let \(X\) and \(X'\) denote a single set under two topologies \(\mathcal{T}\) and \(\mathcal{T}'\), respectively; assume that \(\mathcal{T}' \supset \mathcal{T}\). If one of the spaces is Hausdorff (or regular, or normal), what does that imply about the other?
Solution
Goal: Determine which separation properties (Hausdorff, regular, normal) are preserved when passing to a finer topology (\(\mathcal{T} \subset \mathcal{T}'\)) or a coarser topology (\(\mathcal{T}' \supset \mathcal{T}\)), providing proofs and counterexamples.
1. Hausdorff property (\(T_2\)):- If \((X, \mathcal{T})\) is Hausdorff, then \((X, \mathcal{T}')\) is Hausdorff.- If \((X, \mathcal{T}')\) is Hausdorff, \((X, \mathcal{T})\) need not be Hausdorff.Proof:
1.1. Forward preservation: Suppose \((X, \mathcal{T})\) is Hausdorff. For any distinct \(x \neq y \in X\), there exist disjoint open sets \(U, V \in \mathcal{T}\) with \(x \in U\) and \(y \in V\). Since \(\mathcal{T} \subset \mathcal{T}'\), \(U, V \in \mathcal{T}'\), so \((X, \mathcal{T}')\) is Hausdorff.
1.2. Backward failure: The discrete topology \(\mathcal{T}'\) on an infinite set \(X\) is Hausdorff, but the coarser cofinite topology \(\mathcal{T}\) on \(X\) is not Hausdorff.
2. Regular property (\(T_3\)):Neither direction holds:- \((X, \mathcal{T})\) regular does NOT imply \((X, \mathcal{T}')\) is regular.- \((X, \mathcal{T}')\) regular does NOT imply \((X, \mathcal{T})\) is regular.Proof:
2.1. Forward failure: The standard Euclidean topology \(\mathcal{T}\) on \(\mathbb{R}\) is regular (and metric). The \(K\)-topology \(\mathcal{T}' = \mathcal{T}_K\) is strictly finer than \(\mathcal{T}\), but \((\mathbb{R}, \mathcal{T}_K)\) is not regular because the closed set \(K = \{1/n \mid n \in \mathbb{Z}_+\}\) and the point \(0 \notin K\) cannot be separated by disjoint open sets in \(\mathcal{T}_K\).
2.2. Backward failure: The discrete topology \(\mathcal{T}'\) on an infinite set \(X\) is regular, but the coarser cofinite topology \(\mathcal{T}\) is not regular.
3. Normal property (\(T_4\)):Neither direction holds:- \((X, \mathcal{T})\) normal does NOT imply \((X, \mathcal{T}')\) is normal.- \((X, \mathcal{T}')\) normal does NOT imply \((X, \mathcal{T})\) is normal.Proof:
3.1. Forward failure: The standard Euclidean topology \(\mathcal{T}\) on \(\mathbb{R}\) is normal, but the strictly finer \(K\)-topology \(\mathcal{T}_K\) is not even regular, and therefore not normal.
3.2. Backward failure: The discrete topology \(\mathcal{T}'\) on an infinite set \(X\) is normal, but the coarser cofinite topology \(\mathcal{T}\) is not normal (it is not even Hausdorff).
4. Summary Table of Implications:- Hausdorff: Coarser \(\implies\) Finer (yes); Finer \(\implies\) Coarser (no).- Regular: Coarser \(\implies\) Finer (no); Finer \(\implies\) Coarser (no).- Normal: Coarser \(\implies\) Finer (no); Finer \(\implies\) Coarser (no).Q.E.D.