The Hausdorff condition is an essential part of the definition of a manifold; it is not implied by the other parts of the definition. Consider the following space. Let \(X\) be the union of the set \(\mathbb{R} - \ts{0}\) and the two-point set \(\ts{p, q}\). Topologize \(X\) by taking as basis the collection of all open intervals in \(\mathbb{R}\) that do not contain 0, along with all sets of the form \((-a, 0) \cup \ts{p} \cup (0, a)\) and all sets of the form \((-a, 0) \cup \ts{q} \cup (0, a)\), for \(a > 0\). The space \(X\) is called the line with two origins.
Check that this is a basis for a topology.
Show that each of the spaces \(X - \ts{p}\) and \(X - \ts{q}\) is homeomorphic to \(\mathbb{R}\).
Show that \(X\) satisfies the \(T_1\) axiom, but is not Hausdorff.
Show that \(X\) satisfies all the conditions for a 1-manifold except for the Hausdorff condition.
Solution
Goal: Prove the topological properties of the line with two origins \(X = (\mathbb{R} \setminus \{0\}) \cup \{p, q\}\).
1. Part (a): Basis verification.Proof:
1.1. Covering: For any \(x \in \mathbb{R} \setminus \{0\}\), \(x \in (x - \frac{|x|}{2}, x + \frac{|x|}{2}) \subset \mathbb{R} \setminus \{0\}\). For \(p\) and \(q\), \(p \in (-1, 0) \cup \{p\} \cup (0, 1)\) and \(q \in (-1, 0) \cup \{q\} \cup (0, 1)\).
1.2. Intersections:- The intersection of two intervals in \(\mathbb{R} \setminus \{0\}\) is an interval in \(\mathbb{R} \setminus \{0\}\) or empty.- The intersection of an interval in \(\mathbb{R} \setminus \{0\}\) with a set \(U_a(p) = (-a, 0) \cup \{p\} \cup (0, a)\) is open in \(\mathbb{R} \setminus \{0\}\).- \(U_a(p) \cap U_b(p) = U_{\min\{a, b\}}(p)\), and similarly \(U_a(q) \cap U_b(q) = U_{\min\{a, b\}}(q)\).- \(U_a(p) \cap U_b(q) = (-\min\{a, b\}, 0) \cup (0, \min\{a, b\})\), which is a union of two basic open intervals in \(\mathbb{R} \setminus \{0\}\).
1.3. Hence the collection forms a basis for a topology on \(X\).
2. Part (b): Homeomorphisms with \(\mathbb{R}\).Proof:
2.1. Define \(f: X \setminus \{q\} \to \mathbb{R}\) by \(f(x) = x\) for \(x \in \mathbb{R} \setminus \{0\}\) and \(f(p) = 0\).
2.2. \(f\) is a bijection.
2.3. \(f\) maps basic intervals \((a, b) \subset \mathbb{R} \setminus \{0\}\) to the identical intervals in \(\mathbb{R}\), and maps \(U_a(p)\) to \((-a, a)\).
2.4. Since open intervals not containing \(0\) together with the symmetric intervals \((-a, a)\) generate the standard Euclidean topology on \(\mathbb{R}\), \(f\) is a homeomorphism.
2.5. By the identical argument with \(g(x) = x\) for \(x \neq 0\) and \(g(q) = 0\), \(X \setminus \{p\} \cong \mathbb{R}\).
3. Part (c): \(X\) is \(T_1\) but not Hausdorff.Proof:
3.1. \(T_1\) axiom: For any \(x \in \mathbb{R} \setminus \{0\}\), \(\{x\}\) is closed because its complement is open. For \(p\), \(X \setminus \{p\} = X \setminus \{q\} \setminus \{p\} \cup (X \setminus \{p\})\) is open (since \(X \setminus \{p\} \cong \mathbb{R}\) is open in \(X\)), so \(\{p\}\) is closed. Similarly \(\{q\}\) is closed. Thus \(X\) is \(T_1\).
3.2. Not Hausdorff: Let \(U\) be any open neighborhood of \(p\) and \(V\) any open neighborhood of \(q\).
3.3. \(U\) contains a basic neighborhood \(U_a(p)\) and \(V\) contains a basic neighborhood \(U_b(q)\) for some \(a, b > 0\).
3.4. The intersection \(U \cap V \supseteq U_a(p) \cap U_b(q) = (-\min\{a, b\}, 0) \cup (0, \min\{a, b\}) \neq \emptyset\).
3.5. Thus \(p\) and \(q\) cannot be separated by disjoint open sets, so \(X\) is not Hausdorff.
4. Part (d): \(X\) is a non-Hausdorff 1-manifold.Proof:
4.1. Locally 1-Euclidean: By 2, every point \(x \in X \setminus \{q\}\) has the open neighborhood \(X \setminus \{q\} \cong \mathbb{R}\), and \(q\) has the open neighborhood \(X \setminus \{p\} \cong \mathbb{R}\). Thus every point has a neighborhood homeomorphic to \(\mathbb{R}^1\).
4.2. Second-countable: The countable collection of intervals \((a, b)\) with \(a, b \in \mathbb{Q}\) and \(0 \notin [a, b]\), together with \(U_{1/n}(p)\) and \(U_{1/n}(q)\) for \(n \in \mathbb{Z}_+\), forms a countable basis for \(X\).
4.3. Therefore \(X\) is locally 1-Euclidean and second-countable, satisfying all axioms of a 1-manifold except the Hausdorff condition. Q.E.D.