METRIC TOPOLOGY: A FIRST COURSE
... metric space and continuous function. This follows a familiar pattern in modern pure mathematics: one studies certain structured sets, along with “structurerespecting” functions between them. For example, in linear algebra the basic concepts are those of vector space and linear transformation. In th ...
... metric space and continuous function. This follows a familiar pattern in modern pure mathematics: one studies certain structured sets, along with “structurerespecting” functions between them. For example, in linear algebra the basic concepts are those of vector space and linear transformation. In th ...
On Upper and Lower faintly I-continuous Multifunctions
... S of an ideal topological space (X, τ , I) is said to be I-open [5] if S ⊂ Z(S ∗ ). The complement of an I-closed set is said to be an I-open set. The I-closure [8] and the I-interior [6], that can be defined in the same way as (A) and Z(A), respectively, will be denoted by I(A) and IZ(A), respectiv ...
... S of an ideal topological space (X, τ , I) is said to be I-open [5] if S ⊂ Z(S ∗ ). The complement of an I-closed set is said to be an I-open set. The I-closure [8] and the I-interior [6], that can be defined in the same way as (A) and Z(A), respectively, will be denoted by I(A) and IZ(A), respectiv ...
A study on compactness in metric spaces and topological spaces
... Theorem 2.3 : Any closed subspace of a compact space is compact. Proof: Let Y be a closed of a compact space X, and let {Gi} be an open cover of Y. Each Gi, being open in the relative topology on Y, is the intersection with Y of an open subset Hi of X. Since Y is closed, the class composed of Y′ and ...
... Theorem 2.3 : Any closed subspace of a compact space is compact. Proof: Let Y be a closed of a compact space X, and let {Gi} be an open cover of Y. Each Gi, being open in the relative topology on Y, is the intersection with Y of an open subset Hi of X. Since Y is closed, the class composed of Y′ and ...
Geometric homology versus group homology - Math-UMN
... path from x0 to it, α specifies a class we’ll denote [α] in π1 (X 1 , x0 ). Existence of a homotopy of F ◦ α to a constant map in Y is (of course) equivalent to F∗ ◦ j∗ ([α]) being 1 in π1 (Y, y0 ). Already j∗ ([α]) is 1 in π1 (X, x0 ), so its image under the group homomorphism f is certainly 1 in π ...
... path from x0 to it, α specifies a class we’ll denote [α] in π1 (X 1 , x0 ). Existence of a homotopy of F ◦ α to a constant map in Y is (of course) equivalent to F∗ ◦ j∗ ([α]) being 1 in π1 (Y, y0 ). Already j∗ ([α]) is 1 in π1 (X, x0 ), so its image under the group homomorphism f is certainly 1 in π ...
Transreal calculus - CentAUR
... theory and integration theory. Specifically we note that {−∞}, {∞} and {Φ} are singleton sets that are not finitely path connected to any other numbers. This retains compatibility with an older view of the topology of the transreal numbers, based on computing -neighbourhoods using transreal arithme ...
... theory and integration theory. Specifically we note that {−∞}, {∞} and {Φ} are singleton sets that are not finitely path connected to any other numbers. This retains compatibility with an older view of the topology of the transreal numbers, based on computing -neighbourhoods using transreal arithme ...
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.