목록General topology 10
strangerRidingCaml
Introduction to Homotopy Theory Fundamental Group and Homotopy Fundamental Group: The fundamental group \π1(X,x0 \) of a topological space \( X \) with basepoint \( x_0 \) is a group that captures information about the possible ways loops in \( X \) based at \( x_0 \) can be continuously deformed to each other. Homotopy: Two continuous maps \( f, g: X \righta..
Complete Spaces and Uniform Spaces Completeness in Metric Spaces Completeness: A metric space \(X,d \) is said to be complete if every Cauchy sequence in \X\)convergestoalimitin\(X\).Thatis,foreverysequence\((xn \) in \X\),ifforevery\(ε>0\)thereexistsan\(N\)suchthatforall\(m,n>N\),\(d(xm,xn Examples: ..
Product Spaces Product Topology and Its Properties Product Topology: Let \(Xi,τi \) be a collection of topological spaces indexed by \( i \) from some index set \( I \). The product topology on the Cartesian product \( \prod_{i \in I} X_i \) is defined to be the topology generated by the basis consisting of all sets of the form \( \prod_{i \in I} U_i \), where \( U_i \) is..
Connectedness Definition and Properties of Connected Spaces Connected Space: A topological space \X\)issaidtobeconnectedifitcannotbedividedintotwodisjointnonemptyopensets.Inotherwords,therearenonontrivialclopensets(setsthatarebothopenandclosed in \( X \). Properties of Connected Spaces: If \( X \) is connected, t..
Compactness and Compactification Definition and Properties of Compact Spaces Compact Space: A topological space \( X \) is said to be compact if every open cover of \( X \) has a finite subcover. Equivalently, \( X \) is compact if every open cover of \( X \) can be reduced to a finite subcover. Properties of Compact Spaces: Every closed subset of ..
Separation Axioms T0, T1, T2, and T3 Separation Axioms T0 Separation Axiom: A topological space \( X \) satisfies the T0 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exists an open set containing exactly one of them. T1 Separation Axiom: A topological space \( X \) satisfies the T1 separation axiom if for every pair of distinct p..