Solve EACH of the exercises 1-3
... (a) Show that if f is continuous, then its graph G(f ) is a closed subset of X ×Y. (b) Show that if f, g: [0, 1] → [0, 1] are continuous functions, then dist(G(f ), G(g)) = 0 if, and only if, f (x) = g(x) for some x ∈ X. Note: The interval [0, 1] and its square [0, 1]2 are considered with the standa ...
... (a) Show that if f is continuous, then its graph G(f ) is a closed subset of X ×Y. (b) Show that if f, g: [0, 1] → [0, 1] are continuous functions, then dist(G(f ), G(g)) = 0 if, and only if, f (x) = g(x) for some x ∈ X. Note: The interval [0, 1] and its square [0, 1]2 are considered with the standa ...
Topology - Homework Sets 8 and 9
... Proof 2: Suppose Q × {0} is closed. Then it is the zero set of a finite number of polynomials p1 , . . . , pK . Note that (π, 0) is not in Q × {0}. However, there is a sequence ( xn )n of rational numbers with the property that xn converges to π in the standard topology (e.g., take xn to be the n p ...
... Proof 2: Suppose Q × {0} is closed. Then it is the zero set of a finite number of polynomials p1 , . . . , pK . Note that (π, 0) is not in Q × {0}. However, there is a sequence ( xn )n of rational numbers with the property that xn converges to π in the standard topology (e.g., take xn to be the n p ...
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Logic – Homework 4
... • A more convenient way of representing a topology T is (sometimes) by means of a base: S S For any collection C ⊆ 2X of subsets of X, let T(C) := { O∈C O | C ⊆ T } (with ∅ = O∈∅ O). If T(C) is a topology, we call C a base of T(C) and say that C generates the topology T(C). Every topology T has a ba ...
... • A more convenient way of representing a topology T is (sometimes) by means of a base: S S For any collection C ⊆ 2X of subsets of X, let T(C) := { O∈C O | C ⊆ T } (with ∅ = O∈∅ O). If T(C) is a topology, we call C a base of T(C) and say that C generates the topology T(C). Every topology T has a ba ...
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THE UNIVERSITY OF TOLEDO Topology M.A. Comprehensive Examination L. Bentley H. Wolff
... 5. Prove that every continuous mapping of a compact space into a Hausdorff space is closed. ...
... 5. Prove that every continuous mapping of a compact space into a Hausdorff space is closed. ...
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... is an open subset of X. The space X ∗ is called the quotient space of the space X with respect to ∼. It is often written X/ ∼. The projection map π : X −→ X ∗ which sends each element of X to its equivalence class is always a continuous map. In fact, the map π satisfies the stronger property that a ...
... is an open subset of X. The space X ∗ is called the quotient space of the space X with respect to ∼. It is often written X/ ∼. The projection map π : X −→ X ∗ which sends each element of X to its equivalence class is always a continuous map. In fact, the map π satisfies the stronger property that a ...
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.