Connectedness intuitively, a space is connected if it is all in one piece. This document contains some exercises in algebraic topology, category theory, and homological algebra. But there are connected spaces which are not path connected. Topology of lie groups lecture 1 indian institute of. A basis b for a topology on xis a collection of subsets of xsuch that 1for each x2x. We can generalize the above proof to n subsets, but lets use induction to prove it. X t x of all these tshaped spaces is connected because it is a union of connected spaces such that x. Show that xis pathconnected and connected, but not locally connected or locally pathconnected. The proof combines this with the idea of pulling back the partition from the given topological space to. A topological space is said to be connected if it cannot be represented as the union of two disjoint, nonempty, open sets. The points fx that are not in o are therefore not in c,d so they remain at least a.
A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed specifically, a homotopy of paths, or pathhomotopy, in x is a family of paths f t. Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. The topologists sine curve we want to present the classic example of a space which is connected but not pathconnected. But isnt locally pathconnected so pretty much any standard tools in algebraic topology arent going to help you out. If is pathconnected under a topology, it remains pathconnected when we pass to a coarser. Metrics may be complicated, while the topology may be simple can study families of metrics on a xed topological space ii. A pathcomponent of x is an equivalence class of x under the equivalence relation which makes x equivalent to y if there is a path from x to y.
More precisely, any path connected space is connected. The idea of associating algebraic objects or structures with topological spaces arose early in the history of topology. A set x with a topology tis called a topological space. Here is a typical way these connectedness ideas are used.
Roughly speaking, a connected topological space is one that is \in one piece. But ill leave it here and let you study this problem on yourself. X y is a continuous map between topological spaces, x is connected. While this definition is rather elegant and general, if is connected, it does not imply that a path exists between any pair of points in thanks to crazy examples like the topologists sine curve. A topological space xis path connected if for every x. An example of such spaces is the closure of the graph of sin1x. The key fact used in the proof is the fact that the interval is connected. Since o was assumed to be open, there is an interval c,d about fx0 that is contained in o. Click download or read online button to get topology book now. Let x be path connected, locally path connected, and semilocally simply connected. The book may also be used as a supplementary text for courses in general or pointset topology so that students will acquire a lot of concrete examples of spaces and maps.
Along with the standard pointset topologytopicsconnected and pathconnected spaces, compact spaces. As usual, we use the standard metric in and the subspace topology. Connectedness is one of the principal topological properties that is used to distinguish topolog ical spaces. Prove that a connected open subset xof rnis pathconnected using the following steps. Let x be a pathconnected topological space and consider some x 0. Basic pointset topology 3 means that fx is not in o. Show that xis locally path connected and locally connected, but is not path connected or connected. Assuming such an fexists, we will deduce a contradiction.
Obviously, the integers are connected in the cofinite topology, but to prove that they are not path connected is much more subtle. Then bis a basis of a topology and the topology generated by bis called the standard topology of r. A prerequisite for the course is an introductory course in real analysis. Most of them can be found as chapter exercises in hatchers book on algebraic topology. Homework 2 mth 869 algebraic topology joshua ruiter may 3, 2020 proposition 0. Topologyconnectedness wikibooks, open books for an open world. If is path connected under a topology, it remains path connected when we pass to a coarser topology than. Show that xis path connected and connected, but not locally connected or locally path connected. The locally pathconnected coreflection part i wild topology. The basic assumption is that the participants are familiar with the algebra of lie group theory. Since both parts of the topologists sine curve are themselves connected, neither can be partitioned into two open sets. A metric space is a set x where we have a notion of distance. Roughly speaking, a connected topological space is one that is in one piece.
Find all di erent topologies up to a homeomorphism on a set consisting of 4 elements which make it a connected topological space. The basic incentive in this regard was to find topological invariants associated with different structures. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed. However, it is true that connected and locally pathconnected implies pathconnected. A topological space for which there exists a path connecting any two points is said to be pathconnected. This paper contains a general study of the topological properties of path component spaces including their relationship to the zeroth dimensional. A stronger notion is that of a pathconnected space. Prove that ux is open, by showing for each y2ux there is a 0 such that b y. We will also explore a stronger property called pathconnectedness. Spaces that are connected but not path connected keith conrad.
Its connected components are singletons,whicharenotopen. The pro nite topology on the group z of integers is the weakest topology. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. This site is like a library, use search box in the widget to get ebook that you want. This post is about a simple but remarkably useful construction that will give you a locally pathconnected spaces which has the same. Feb 16, 2015 now let us discuss the topologists sine curve. A topology on a set x is a collection tof subsets of x such that t1. Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence.
If two topological spaces are connected, then their product space is also connected. A path from a point x to a point y in a topological space x is a continuous function. A topological space is said to be path connected if for any two points. More speci cally, we will show that there is no continuous function f. The most fundamental example of a connected set is the interval 0. If is pathconnected under a topology, it remains pathconnected when we pass to a coarser topology than. A pathconnected space is a stronger notion of connectedness, requiring the structure of a path. Any space may be broken up into pathconnected components. If is a path connected space and is the image of under a continuous map, then is also path connected. If is a pathconnected space and is the image of under a continuous map, then is also pathconnected. I admit that this looks like the next best homework problem and was dismissed as such in this thread, but if you think about it, it does not seem to be obvious at all. Careful, this is not the set of all points with both coordinates irrational. So far i can picture, i think they should be equivalent. Geometrically, the graph of y sin1x is a wiggly path that.
Exam i solutions algebraic topology october 19, 2006 1. Topology of lie groups lecture 1 in this seminar talks, to begin with, we plan to present some of the classical results on the topology of lie groups and homogeneous spaces. The locally pathconnected coreflection part i wild. Contents the fundamental group the university of chicago. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. Homotop y equi valence is a weak er relation than topological equi valence, i. Let us introduce an equivalence relation on x by x0. A topological space x is pathconnected if any two points in x can be joined by a continuous path. A topological space is called if, for every pair of points. Mathematics 490 introduction to topology winter 2007 what is this. X be covering spaces corresponding to the subgroups hi. Proof let x be a pathconnected topological space, and let f. Obviously, the integers are connected in the cofinite topology, but to prove that they are not pathconnected is much more subtle.
Let r 2 be the set of all ordered pairs of real numbers, i. Sis not path connected now that we have proven sto be connected, we prove it is not path connected. Connected subsets of the real line are either onepoint sets or intervals. The set of pathconnected components of a space x is often denoted. Some interesting topologies do not come from metrics zariski topology on algebraic varieties algebra and geometry the weak topology on hilbert space analysis any interesting topology on a nite set combinatorics 2 set. A subset of a topological space is called connected if it is connected in the subspace topology. A topological space x is pathconnected if every pair of points is connected by a path. Say youve got some pathconnected space and you want to know about its fundamental group. Suppose that there are two nonempty open disjoint sets a and b whose union is x 1. Let x be a path connected topological space and consider some x 0. Specifically, a homotopy of paths, or pathhomotopy, in x is a family of paths ft. The space xis said to be locally path connected if for each x.
The goal of this part of the book is to teach the language of mathematics. Why are the integers with the cofinite topology not path. If a topological space xis contractible, then it is pathconnected. Consider the intersection eof all open and closed subsets of x containing x. And any open set which contains points of the line segment x 1 must contain points of x 2. Oct 11, 2014 say youve got some pathconnected space and you want to know about its fundamental group. However, it is true that connected and locally path connected implies path connected. The topology of path component spaces jeremy brazas october 26, 2012 abstract the path component space of a topological space x is the quotient space of x whose points are the path components of x. A topological space xis pathconnected if for every x. Show that xis locally pathconnected and locally connected, but is not pathconnected or connected. A separation of xis a pair of disjoint nonempty open sets uand v in xwhose union is x. Along with the standard pointset topology topicsconnected and pathconnected spaces, compact spaces, separation axioms, and metric spacestopology covers the construction of spaces from other spaces, including products and quotient spaces.
Note that the cocountable topology is ner than the co nite topology. A topological space xis path connected if to every pair of points x0,x1. X be the connected component of xpassing through x. The simplest example is the euler characteristic, which is a number associated with a surface. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. A topological space x is path connected if any two points in x can be joined by a continuous path. The space x is connected if there does not exist a separation of x.