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| Question | Answer |
| Open set | A subset A of \(ℝ^2 \) is called \(\underline {open} \) if each point p ∈ A has a ball around it included in A: ∀ p ∈ A ∃ ε > 0 s.t. B(p,ε) ⊆ A. |
| Properties of Open Sets in \(ℝ^2 \) | - \(ℝ^2 \) is open - ∅ is open (where ∅ = empty-set) - if A & B are open, so is A∩B - if {\(A_i\): i ∈ I} is a collection of open sets, then their union \(\underset{i∈I}{∪}\)\(A_i\) is open |
| Closure Points | Point p is \(\underline {close} \) to A or is a \(\underline {closure-point-of-A} \) if every open ball around p intersects A: ∀ ε > 0, B(p,ε)∩A ≠ ∅ |
| Theorem about \(ℝ^2 \) | p is close to A iff every \(\underline{open-set}\) containing p intersects A. |
| Closed Set | Set A is \(\underline {closed} \) if every point that is close to A actually belongs to A i.e. if A contains its closure points |
| Theorem in \(R^2 \) | A is closed iff (\(R^2 \) -A), it's complement, is open. (Note: also works conversely) |
| Topology | A \(\underline {topology} \) on a set is a collection T of subsets a of X. |
| Topological Space | Is a pair (X, T), with T a topology of set X |
| Comparing Topologies | If \(T_1\) & \(T_2\) are topologies on X & \(T_1\) ⊆ \(T_2\). We say \(T_2\) is finer/stronger/larger then \(T_1\); and \(T_1\) is coarser/weaker/smaller than \(T_2\). |
| Neighbourhood (nbhd) | A \(\underline {nbhd} \) of a point x in a top. space is any open set U containing x i.e. any U ∈ T with x ∈ U |
| Theorem 1.4: open subset | In any space (X,T) a subset A is \(\underline {open} \) iff every point of A has a nbhd included in A. |
| Closed sets | In any top. space (X,T) a set A ⊆ X is defined to be \(\underline {closed} \) iff its complement is open i.e. A is closed iff (X-A) ∈ T |
| Theorem 1.17: In any top. space X... | In any top. space X: - ∅ & X are closed. - If A & B are closed, so is A∪B - If {\(C_i \): i ∈ I} is a collection of closed sets, the \(\underset{i∈I}{∩}\)\(C_i \) is closed. |
| Definition 1.18: Hausdorff Space | X is \(\underline {Hausdorff} \) if every pair of disjoint points x,y ∈ X ∃ disjoint nbhd's U & V of x & y respectively. i.e. if x≠y then there are open sets U & V with x ∈ U, y ∈ V & U∩V=∅ |
| Theorem 1.19: singleton in Hausdorff space | In any Hausdorff space any singleton is closed. Hence any \(\underline {finite-set} \) is closed. |
| Definition 1.5: Basis | If X is any set, a collection \(\underline {B} \) of subsets of X is called a \(\underline {basis} \) (for a topology) on X if: (i) For each number of X belongs to some number of \(\underline {B} \). i.e. ∀ x∈X ∃ B∈\(\underline {B} \) s.t. x∈B. Thus ∪\(\underline {B} \)=X (ii) For any \(B_1\),\(B_2\) ∈ \(\underline {B} \) & any x∈\(B_1\)∩\(B_2\) there is a \(B_3\)∈ \(\underline {B} \) with x∈\(B_3\) ⊆\(B_1\)∩\(B_2\). |
| Definition 1.6: Topology \(\underline {T} \) generated by \(\underline {B} \) | If \(\underline {B} \) is a basis on set X, the topology \(T_B \) genersted by \(\underline {B} \) is defined by: For any U⊆X, U∈\(T_B\) iff (i) U = ∅; or, (ii) U is a union of a collection of basis members |
| Theorem 1.8 | If \(\underline {B} \) is a basis on X, then \(T_B\) IS a topology on X |
| Theorem 1.9 | If \(\underline {B} \) is a basis on X, then U∈\(T_B\) iff ∀x∈U ∃\(B_x\)∈\(\underline {B} \) s.t. x∈\(B_x\)⊆U i.e. U = \(\underset{x∈U}{∪}\)\(B_x\) |
| Theorem 1.13: Collection of open sets | Let \(\underline {C} \) ce a collection of open sets in space (X,T) (i.e.\(\underline {C} \)⊆T). Then \(\underline {C} \) is a basis for T (i.e.\(T_C\)=T) iff every open U∈T is a union of members of \(\underline {C} \) i.e. iff ∀u∈T, ∀x∈U ∃V∈\(\underline {C} \) s.t. x∈V⊆U. |
| Theorem: Comparing Topologies via Bases | Let X have topologies T & T', with bases \(\underline {B} \) & \(\underline {B'} \) respectively. Then T' is finer than T (i.e. T⊆T') iff every B∈\(\underline {B} \) is a union of members \(\underline {B'} \). |
| Subbasis | Let (X,T) be a top. space & \(\underline {S}\) be a collection of open sets in that space (i.e. \(\underline {S}\)⊆T). Then \(\underline {S}\) is a \(\underline {subbasis}\) for T, which means that \(T_S\)=T, iff \(B_S\) is a basis for T iff every T-open set is a union of members of \(B_S\). |
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