ENTRY TOPOLOGY Authors: Oliver Knill 2003, John Carlson 2003-2004 Literature: http://at.yorku.ca/cgi-bin/bell/props.cgi +------------------------------------------------------------ | Alexander compactification +------------------------------------------------------------ The Alexander compactification Y of a Hausdorff space (X,O) is the topological space (Y = X cup x,P), where x is an additional point. The topology P consists of the elements in O and the complements of closed subsets as neighborhoods of that point. The new topological space Y is compact. +------------------------------------------------------------ | Alexander's subbase theorem +------------------------------------------------------------ The Alexander's subbase theorem: if every open cover of a topological space X has a finite sub-cover then X is compact. +------------------------------------------------------------ | arc-connected +------------------------------------------------------------ A topological space is called arc-connected if any two points can be connected by a path, a continuous image of an interval. Path connected is stronger than connected but not equivalent: the subset (x,sin(1/x)), x in R^+ cup (0,y), -1 leq y leq 1 of the plane with topology induced from the plane is connected but not path connected. Arc-connected is also called path-connected. +------------------------------------------------------------ | Baire category +------------------------------------------------------------ Baire category is a measure for the size of a set in a topological space. Countable unions of nowhere dense sets are called of the first categorie or meager, any other set of second category. Complements of meager sets are called residual. Baire category is used to quantify certain sets. For example it is known that "most" numbers are Liouville numbers in the sense that they form a residual set among all real numbers. +------------------------------------------------------------ | Baire space +------------------------------------------------------------ A Baire space is a topological space with the property that the intersection of countable family of open dense subsets is dense. +------------------------------------------------------------ | Baire category theorem +------------------------------------------------------------ The Baire category theorem: a complete metric space is a Baire space. +------------------------------------------------------------ | ball +------------------------------------------------------------ A ball in a metric space is a set of the form y | d(x,y) 1 is connected but not locally connected because small neighborhoods of the point (0,1) are not connected. +------------------------------------------------------------ | Hausdorff +------------------------------------------------------------ A topological space (X,T) is called Hausdorff if for every two points x,y in X, there are disjoint open sets U,V in T such that x in U and y in V. This is refined through seperation axioms, T0, ..., T4. Hausdorff is also called T2. Any metric space is Hausdorff: if d is the distance betwen x and y, then balls of radius d/3 around x and y seperate the points. The plane X with semimetric d(x,y) = |x_1-y_1| is not Hausdorff: the points x=(0,-1) and y=(0,1) can not be seperated by open sets. +------------------------------------------------------------ | seperation axioms +------------------------------------------------------------ seperation axioms define classes of topological spaces with decreasing seperability properties: T4 Rightarrow T3 Rightarrow T2 Rightarrow T1 Rightarrow T0. T0 space: for two different points x,y in X one of the points has an open neighborhood U not containing the other point. T1 space: for two different points x,y in X there exists an open neighborhood U of x and an open neighborhood V of y. such that x is not in V and y is not in U. T2 space: also called Hausdorff" two different points x,y can be seperated with disjoint neighborhoods U,V. T3 space: T1 and regular: any point x and any closed set F not containing x can be seperated by two disjoint neighborhood. T4 space: T1 and normal: any two disjoint sets F,G can be separated by two disjoint open sets. It is known that a T4 space with a countable basis is metrizable. +------------------------------------------------------------ | Hausdorff topology +------------------------------------------------------------ The Hausdorff topology is a metric on the set of closed bounded subsets of a complete metric space. The distance between two sets A and B is the infimum over all r for which A is contained in a r-neighborhood of B and B is contained in a r-neighborhood of A. +------------------------------------------------------------ | Lindeloef +------------------------------------------------------------ A topological space is called Lindeloef if every open cover of X contains a countable subcover. +------------------------------------------------------------ | compact +------------------------------------------------------------ A topological space is called compact if every open cover of X contains a finite subcover. Examples of results known about compactnes: Heine-Borel theorem: a closed interval in the real line is compact. If f: X to Y is continuous and onto and X is compact, then Y is compact. As a consequence, a continouous function on a compact subspace has both a maximum and a minimum. In a Hausdorff space, compact sets are closed. In a metric space, compact sets are closed and bounded. Closed subsets of compact spaces are compact. Tychonof theorem: the product of a collection of compact spaces is compact. +------------------------------------------------------------ | countably compact +------------------------------------------------------------ A topological space is called countably compact if every countable open cover of X contains a finite subcover. +------------------------------------------------------------ | locally compact +------------------------------------------------------------ A topological space is called locally compact if every point has a neighborhood, which has a compact closure. Examples. The real line is compact but not locally compact. A compact Hausdorff space is locally compact. The n-dimensional Euclidean space R^n is lcally compact but not compact. +------------------------------------------------------------ | locally compact +------------------------------------------------------------ A set U of open sets in a topological space (X,O) is called locally finite if every point x in X has a neighborhood V, such that V has a nonempty intersection with only finitely many elements in U. +------------------------------------------------------------ | paracompact +------------------------------------------------------------ A topological space (X,O) is called paracompact if every open cover has a countable, locally finite subcover. +------------------------------------------------------------ | relatively compact +------------------------------------------------------------ A subset A of a topological space (X,T) is called relatively compact if the closure of A is compact. +------------------------------------------------------------ | filter +------------------------------------------------------------ A filter on a nonempty set X is a set of subsets F satisfying X is in F, but the empty set emptyset is not in F. If A and B are in F, then their intersection is in F. If A is in F and B is a subset of A, then B is in F. Examples: Principal filter for a nonempty subset A consists of all subsets of X which contain A. Frechet filter for an infinite set consists of all subsets of X such that their complement is finite. Neighborhood filter of a point x in a topological space (X,T) is the set of open neighborhoods of x. Elementary filter for a sequence x_n in X consists of all sets A in X such that x_n is in A for large enough n. +------------------------------------------------------------ | converges +------------------------------------------------------------ A sequence x_n in a topological space converges to a point x, if for every neighborhood U of x, there exists an integer m, such that for n>m one has x_n in U. +------------------------------------------------------------ | Filter convergence +------------------------------------------------------------ Filter convergence A filter F converges to x in a topological space (X,T) if F contains the neighborhood filter G of x, that is if F contains all neighborhoods of x. For example, an elementary filter to a sequence x_n converges to a point x, if and only if x_n converges to x. +------------------------------------------------------------ | accumulation point +------------------------------------------------------------ A point y is called an accumulation point of a filter F, if there exists a filter G containing F such that G converges to x. +------------------------------------------------------------ | directed +------------------------------------------------------------ A set M is called directed if there exists a partial order (M,<) on M satisfying for every two points a,b in M there exists c, with a0, B_r(x)= y | d(x,y) X, where D is a directed set. For example: if D is the set of natural numbers, then a net is a sequence. A net defines a filter F: it is the set of all sets A such that x_t is eventually in A. A net x_t converges to a point x if and only if the associated filter converges to x. +------------------------------------------------------------ | open cover +------------------------------------------------------------ open cover A subset U of O, where (X,O) is a topological space is called an open cover of X if the union of all elements in U is X. If U and V are open covers and V subset U, then V is called a subcover of U. +------------------------------------------------------------ | product space +------------------------------------------------------------ The product space between topological spaces is defined as (X x Y, O x P), where X x Y is the set of all pairs (x,y), x in X, y in Y and O x P is the coarsest topological space which contains all products A x B, where A in O and B in P. For example, if (X,O)=(Y,P) are both the real line with the topology generated by d(x,y) = |x-y|, then the product space is homeomorphic to the plane with the metric d(x,y) = sqrt(x_1-x_2)^2+(y_1-y_2)^2. +------------------------------------------------------------ | second countable +------------------------------------------------------------ A topological space is called second countable, if it has a countable basis. Example. Every seperable metric space is second countable. Especially, every finite-dimensional Euclidean space is second countable. +------------------------------------------------------------ | metrizable +------------------------------------------------------------ A topological space is called metrizable if there exists a metric d on the set X that induces the topology of X. Any regular space with a countable basis is metrizable. +------------------------------------------------------------ | homotopic +------------------------------------------------------------ homotopic If f and g are continuous maps from the topological space X to a topological space Y, we say that f is homotopic to g if there is a continuous map F from X x I to Y, such that F(x,0) = f(x) and F(x,1) = g(x) for all x. For example, the maps f(x) = x^2 and g(x) = sin(x) on the real line are homotopic, because we can define F(x,t) = (1-t)x^2 + t sin(x). The maps f(x) = x and g(x) = sin(2 pi x) on the circle are not homotopic. While g is homotopic to the constant function h(x)=0, the map f(x) can not be deformed to a constant without breaking continuity. +------------------------------------------------------------ | induced topology +------------------------------------------------------------ The induced topology on a subset A of X, where (X,T) is a topological spoace is the the topological space (A, Y cap A _Y in T). +------------------------------------------------------------ | path homotopic +------------------------------------------------------------ path homotopic If f and g are and continuous homotopic maps from an interval to a space X, we say f and g are path homotopic if their images have the same end points. For instance, the maps f(x) = x^2 and g(x) = x^3 are path homotopic on the closed interval from 0 to 1. The maps f(x)=2 x^2 and g(x)=x^3 are homotopic on the unit interval but not path homotopic. +------------------------------------------------------------ | loop +------------------------------------------------------------ A loop is a path in a topological space that begins and ends at the same point. A loop is also called a closed curve. Loops play a role in definitions like simply connected: a topological space is simply connected if every loop is homotopic to a constant loop which is a fancy way telling that every closed path can be collapsed inside X to a point. +------------------------------------------------------------ | fundamental group +------------------------------------------------------------ The fundamental group of a topological space at a point is the set of homotopy classes of loops based at that point. +------------------------------------------------------------ | Topologist's Sine Curve +------------------------------------------------------------ The Topologist's Sine Curve is the union S of the graph of the function sin(1/x) on the positive real axes R^+ with the y-axes. It an example of a topological space which is connected but not path-connected. Proof: if S were path-connected, there would exist a path r(t)=(x(t),y(t)) connecting the two points (0,1) and (0,pi). The set t | r(t) in S is closed. Let T be the largest t in that set for which r(t) is in the y-axes. Then x(T)=0 and r(t)=(x(t),sin(1/x(t)) for t>T. Because there are times t_n >t_n-1>T, t_n to T for which y(t_n)= (-1)^n, the function r(t) can not be continuous at t=T. +------------------------------------------------------------ | Urysohn lemma +------------------------------------------------------------ The Urysohn lemma tells that if X is a normal space and A and B are disjoint closed subsets of X, then there exists a continuous map f from X to the unit interval such that f(x) = 0 for all x in A, and f(x) = 1 for all x in B. Proof: use the normality of X to construct a family U_p of open sets of X indexed by the rational numbers P in the unit interval so that for p