Metric Spaces - Andrew Tulloch
... Theorem 6.8 (Path-connectedness implies connectedness). Let X be a metric space and A a subset of X. If A is path-connected, then A is connected. We note that the converse is not necessarily true - that is, there exist connected sets that are not path connected. However, in Rn , we have the followin ...
... Theorem 6.8 (Path-connectedness implies connectedness). Let X be a metric space and A a subset of X. If A is path-connected, then A is connected. We note that the converse is not necessarily true - that is, there exist connected sets that are not path connected. However, in Rn , we have the followin ...
algebraic topology - School of Mathematics, TIFR
... of addition is an abelian group. Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp ...
... of addition is an abelian group. Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp ...
Decompositions of normality and interrelation among its variants
... continuous functions {fn } defined on A such that |f − i=1 fi | ≤ ( 32 )nPon A. It ...
... continuous functions {fn } defined on A such that |f − i=1 fi | ≤ ( 32 )nPon A. It ...
Ahmet HAMAL and Mehmet TERZILER PERITOPOLOGICAL
... H. CARTAN introduced filters and ultrafilters in 1937. Before that time, it was common practice to consider a topological space as a structure with an idempotent “closure operation”. Right after, (as illustrated by BOURBAKI), a correlative definition for topological spaces bloomed. A topological spa ...
... H. CARTAN introduced filters and ultrafilters in 1937. Before that time, it was common practice to consider a topological space as a structure with an idempotent “closure operation”. Right after, (as illustrated by BOURBAKI), a correlative definition for topological spaces bloomed. A topological spa ...
Norm continuity of weakly continuous mappings into Banach spaces
... quasi-continuous at z0 if, for every open subset U ⊂ X with g(z0 ) ∈ U, there exists some open set V ⊂ Z such that: a) z0 ∈ V (the closure of V in Z); S b) g(V ) := {g(z) : z ∈ V } ⊂ U. The mapping g is called quasi-continuous if it is quasi-continuous at each point of Z. For real-valued functions t ...
... quasi-continuous at z0 if, for every open subset U ⊂ X with g(z0 ) ∈ U, there exists some open set V ⊂ Z such that: a) z0 ∈ V (the closure of V in Z); S b) g(V ) := {g(z) : z ∈ V } ⊂ U. The mapping g is called quasi-continuous if it is quasi-continuous at each point of Z. For real-valued functions t ...
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.