Metric Spaces and Topology M2PM5 - Spring 2011 Solutions Sheet
... this argument works for any family of subsets {Hi }i∈I .) Conversely K, being the union of a finite number of closed sets, is closed. Also, K ⊇ H. Therefore K ⊇ H. So K = H. (iii) Let H = (0, 2) ∪ (3, 4), K = (1, 3). Then H = [0, 2] ∪ [3, 4], K = [1, 3], and H ∩K = (1, 2], H ∩K = [1, 2), H ∩K = [1, ...
... this argument works for any family of subsets {Hi }i∈I .) Conversely K, being the union of a finite number of closed sets, is closed. Also, K ⊇ H. Therefore K ⊇ H. So K = H. (iii) Let H = (0, 2) ∪ (3, 4), K = (1, 3). Then H = [0, 2] ∪ [3, 4], K = [1, 3], and H ∩K = (1, 2], H ∩K = [1, 2), H ∩K = [1, ...
Proof of Lemma 1 from “Brief note on Quotient Spaces” Lemma 1
... Define a function φ : Y → Q = X/ ∼ by φ(y) = π (f −1 (y)). Note that one of the hypotheses is that equivalence classes in X under ∼ are precisely the point-inverses under f , i.e. the point inverses under f are identical to the point inverses under π. Since f −1 (y) is exactly [x] for some x ∈ X, φ( ...
... Define a function φ : Y → Q = X/ ∼ by φ(y) = π (f −1 (y)). Note that one of the hypotheses is that equivalence classes in X under ∼ are precisely the point-inverses under f , i.e. the point inverses under f are identical to the point inverses under π. Since f −1 (y) is exactly [x] for some x ∈ X, φ( ...
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... we prove that U V T . For any x U V , since U BU V = BV , U 1 ,V1 B such that x U 1 V1 , then Wx B such that xWx U1 V1 . Thus U V = xU V Wx . We let BU V Wx , xU V , then U V = BU V. This shows that T is closed under finite intersection. T is a topology. It is easy to see that B ...
... we prove that U V T . For any x U V , since U BU V = BV , U 1 ,V1 B such that x U 1 V1 , then Wx B such that xWx U1 V1 . Thus U V = xU V Wx . We let BU V Wx , xU V , then U V = BU V. This shows that T is closed under finite intersection. T is a topology. It is easy to see that B ...
connected - Maths, NUS
... Theorem 5.4 If Y is a connected subset of a space X then Y is connected. Proof (see page 136 for a different proof). Assume to the contrary that Y is disconnected. Then there exists open subsets A, B X ...
... Theorem 5.4 If Y is a connected subset of a space X then Y is connected. Proof (see page 136 for a different proof). Assume to the contrary that Y is disconnected. Then there exists open subsets A, B X ...
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... A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countabl ...
... A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countabl ...
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... follows for the following reasons: if f : X → Y is a homeomorphism, and g : Y → Z is also a homeomorphism, then: (a) Both f −1 and g −1 are homeomorphisms, since (f −1 )−1 = f and (g −1 )−1 = g. (b) g ◦ f is a bijection since the composition of bijections is a bijection. (c) g ◦ f is continuous, sin ...
... follows for the following reasons: if f : X → Y is a homeomorphism, and g : Y → Z is also a homeomorphism, then: (a) Both f −1 and g −1 are homeomorphisms, since (f −1 )−1 = f and (g −1 )−1 = g. (b) g ◦ f is a bijection since the composition of bijections is a bijection. (c) g ◦ f is continuous, sin ...
The γ-open Open Topology for Function Spaces
... if there exists a semi-open set S such that x ∈ S ⊆ M (X). In [6] Latif introduced the notion of semi-convergence of filters. Let S(x) = {A ∈ SO(X) : x ∈ A} and let Sx = {A ⊆ X : there exists µ ⊆ S(x) such that µ is finite and ∩µ ⊆ A}.Then Sx is called the semi-neighborhood filter at x. For any filt ...
... if there exists a semi-open set S such that x ∈ S ⊆ M (X). In [6] Latif introduced the notion of semi-convergence of filters. Let S(x) = {A ∈ SO(X) : x ∈ A} and let Sx = {A ⊆ X : there exists µ ⊆ S(x) such that µ is finite and ∩µ ⊆ A}.Then Sx is called the semi-neighborhood filter at x. For any filt ...
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... with the diagonal embedding to get back information about X. For instance, X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier to check if Im ∆ is closed than to verify the Hausdorff condition directl ...
... with the diagonal embedding to get back information about X. For instance, X is Hausdorff if and only if the image of ∆ is closed in X × X [proof]. If we know more about the product space than we do about X, it might be easier to check if Im ∆ is closed than to verify the Hausdorff condition directl ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.