Flashcards
§1 Topological Spaces
Cards in this deck (22)
- 1Open setA 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. - 2Properties 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 - 3Closure PointsPoint 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 ≠ ∅ - 4Theorem about \(ℝ^2 \)p is close to A iff every \(\underline{open-set}\) containing p intersects A.
- 5Closed SetSet A is \(\underline {closed} \) if every point that is close to A actually belongs to A
i.e. if A contains its closure points - 6Theorem in \(R^2 \)A is closed iff (\(R^2 \) -A), it's complement, is open.
(Note: also works conversely) - 7TopologyA \(\underline {topology} \) on a set is a collection T of subsets a of X.
- 8Topological SpaceIs a pair (X, T), with T a topology of set X
- 9Comparing TopologiesIf \(T_1\) & \(T_2\) 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 \(T_2\). - 10Neighbourhood (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 - 11Theorem 1.4: open subsetIn any space (X,T) a subset A is \(\underline {open} \) iff every point of A has a nbhd included in A.
- 12Closed setsIn 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 - 13Theorem 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. - 14Definition 1.18: Hausdorff SpaceX 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=∅ - 15Theorem 1.19: singleton in Hausdorff spaceIn any Hausdorff space any singleton is closed. Hence any \(\underline {finite-set} \) is closed.
- 16Definition 1.5: BasisIf 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∈\(B_3\) ⊆\(B_1\)∩\(B_2\). - 17Definition 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 - 18Theorem 1.8If \(\underline {B} \) is a basis on X, then \(T_B\) IS a topology on X
- 19Theorem 1.9If \(\underline {B} \) 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 = \(\underset{x∈U}{∪}\)\(B_x\) - 20Theorem 1.13: Collection of open setsLet \(\underline {C} \) 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. - 21Theorem: Comparing Topologies via BasesLet 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 \(\underline {B'} \). - 22SubbasisLet (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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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Topology (MATH313)) Flashcards on §1 Topological Spaces, created by Amy Wogan on 27/04/2016.
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- Amy Wogan · Apr 2016
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- May 2, 2016
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