DOCSLIB.ORG

2 Separation Axioms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2 Separation Axioms 2 SEPARATION AXIOMS Definition 2.1 A space X is a T0 space iff it satisfies the T0 axiom, i.e. for each x, y ∈ X such that x 6= y there is an open set U ⊂ X so that U contains one of x and y but not the other. Obviously the property T0 is a topological property. An arbitrary product of T0 spaces is T0. Discrete spaces are T0 but indiscrete spaces of more than one point are not T0. Definition 2.2 A space X is a T1 space or Frechet space iff it satisfies the T1 axiom, i.e. for each x, y ∈ X such that x 6= y there is an open set U ⊂ X so that x ∈ U but y∈ / U. T1 is obviously a topological property and is product preserving. Every T1 space is T0. Example 2.3 The set {0, 1} furnished with the topology {∅, {0}, {0, 1}} is called Sierpinski space. It is T0 but not T1. Proposition 2.4 X is a T1 space iff for each x ∈ X, the singleton set {x} is closed. Proof. Easy. Definition 2.5 A space X is a T2 space or Hausdorff space iff it satisfies the T2 axiom, i.e. for each x, y ∈ X such that x 6= y there are open sets U, V ⊂ X so that x ∈ U, y ∈ V and U ∩ V = ∅. T2 is a product preserving topological property. Every T2 space is T1. Example 2.6 Recall the cofinite topology on a set X defined in Section 1, Exercise 3. If X is finite it is merely the discrete topology. In any case X is T1, but if X is infinite then the cofinite topology is not T2. Proposition 2.7 Let f, g : X → Y be maps with Y Hausdorff. Then {x ∈ X / f(x) = g(x)} is closed. Proof. Let A = {x ∈ X / f(x) 6= g(x)}, and suppose x ∈ A. Since f(x) 6= g(x), there are −1 −1 open sets U, V ⊂ Y so that f(x) ∈ U, g(x) ∈ V and U ∩ V = ∅. Let W = f (U) ∩ g (V ). Then W is open and x ∈ W . Moreover, W ⊂ A. Thus A is open, so {x ∈ X / f(x) = g(x)} is closed. In particular if X is T2 and f : X → X is a map then the fixed-point set of f [i.e. the set of points x for which f(x) = x] is closed. Definition 2.8 A space X is regular iff for each x ∈ X and each closed C ⊂ X such that x∈ / C there are open sets U, V ⊂ X so that x ∈ U, C ⊂ V and U ∩ V = ∅. A regular T1 space is called a T3 space. The properties T3 and regular are both topological and product preserving. Every T3 space is T2. 2 Example 2.9 The slit disc topology on R is T2 but not regular, hence not T3. 2 2 2 Take X = R and let P = {(x, y) ∈ R / x 6= 0} and L = {(0, y) ∈ R }. Topologise X as follows (c.f. Theorem 1.12): 2 • if z ∈ P , let the open discs in R centred at z form a basis of neighbourhoods of z; 11 • if z ∈ L let the sets of the form {z} ∪ (P ∩ D) form a basis of neighbourhoods of z where D is an open disc in the plane centred at z. Clearly X is T2. However, note that L is a closed subset of X and that as a subspace, L is discrete. Thus any subset of L is closed in X; in particular, L − {(0, 0)} is closed and does not contain (0, 0), although every open set containing (0, 0) meets every open set containing L − {(0, 0)}. Thus X is not regular and hence not T3. Example 2.10 Every indiscrete space is vacuously regular but no such space (of more than 1 point!) is T0 and hence also no such space is T2. Theorem 2.11 A space X is regular iff for each x ∈ X, the closed neighbourhoods of x form a basis of neighbourhoods of x. Proof. ⇒: given x ∈ X, and a neighbourhood N of x, there is an open set O ⊂ X such that x ∈ O ⊂ N. Consider the point x and the closed set X − O, which does not contain x. By regularity, there are open sets U and V such that x ∈ U, X − O ⊂ V and U ∩ V = ∅. Thus x ∈ U ⊂ X − V ⊂ O ⊂ N, so X − V is a closed neighbourhood of X contained in the given neighbourhood N of x. ⇐: given x ∈ X and the closed set C ⊂ X − {x}, since X − C is open and contains x, there is a closed neighbourhood N of x so that N ⊂ X − C. Let V = X − N. Then V is open and C ⊂ V . Since N is a neighbourhood of x, there is an open set U such that x ∈ U ⊂ N. Then U ∩ V ⊂ N ∩ (X − N) = ∅, so X is regular. Definition 2.12 A space X is normal iff for each pair A, B of disjoint closed subsets of X, there is a pair U, V of disjoint open subsets of X so that A ⊂ U, B ⊂ V and U ∩ V = ∅.A normal T1 space is called a T4 space. The properties T4 and normal are both topological properties but, perhaps surprisingly, are not product preserving. Every T4 space is clearly a T3 space, but it should not be surprising that normal spaces need not be regular. Example 2.13 Sierpinski’s space is vacuously normal but is not regular since 0 and {1} cannot be separated. 2 Example 2.14 The tangent-discs topology on R is T3 but not normal, hence not T4. Let X, P and L be as in Example 2.9 but topologise X as follows: 2 • if z ∈ P , let the open discs in R centred at z form a basis of neighbourhoods of z; • if z ∈ L let the sets of the form {z} ∪ D form a basis of neighbourhoods of z where D is the union of two open discs in P tangent to L at z, one to the left of L and the other to the right of L. Clearly X is T1. X is regular, hence T3. Indeed, suppose z ∈ X and N ∈ N (z). If z ∈ P , then there is an open disc D centred at z so that z ∈ D ⊂ N. The closed disc of half the radius also centred at z is a closed neighbourhood of z and lies in N. If z ∈ L, then there is D as in the definition of the neighbourhood basis of z so that {z} ∪ D ⊂ N. The union of the closed discs of half the radii of those of D also tangent to L at z is a closed neighbourhood of z contained in N. Thus in either case, closed neighbourhoods form a neighbourhood basis so by Theorem 2.11, X is regular. 12 Suppose that X is normal. Let M = {(x, y) / x, y ∈ Q}. As in Example 2.9, every subset of L is a closed subset of X. Thus to each A ⊂ L, we can assign open sets UA and VA of X so that A ⊂ UA, L − A ⊂ VA and UA ∩ VA = ∅. Let MA = M ∩ UA. We show that the function sending A to MA is injective. This will lead to a contradiction because L is uncountable whereas M is countable, so it is impossible to find an injective function from the power set of L to that of M. Loosely speaking, if X were normal then there would need to be more open subsets of X than there are. To show that the function constructed in the last paragraph really is injective, let A, B ⊂ L be such that A 6= B: to show that MA 6= MB. Either A − B 6= ∅ or B − A 6= ∅: assume the former. We have ∅ 6= A − B = A ∩ (L − B) ⊂ UA ∩ VB. Since UB and VB are disjoint and open, we must have UB ∩ VB = ∅, so VB ⊂ X − UB, and hence ∅ 6= UA ∩ VB ⊂ UA − UB. Every non-empty open subset of X meets M so, since UA −UB is open, we have: ∅ 6= M ∩(UA −UB) ⊂ M ∩ (UA − UB) = MA − MB. Thus if A − B 6= ∅ then MA − MB 6= ∅. Similarly, if B − A 6= ∅ then MB − MA 6= ∅, so the function is injective as promised. Example 2.15 Let S be the real line with the right half-open interval topology of Example 1.4. Then S is normal but S × S is not normal. The space S × S is sometimes called Sorgenfrey’s square. Suppose that A and B are disjoint subsets of S. Since B is closed, for each a ∈ A, there 0 0 0 is a > a so that [a, a ) ∩ B = ∅. Let U = ∪a∈A[a, a ). Then U is an open set containing A. 0 0 0 Similarly define V = ∪b∈B[b, b ), where for each b ∈ B, b > b is chosen so that [b, b ) ∩ A = ∅. V too is open and contains B. It is claimed that U ∩ V = ∅, for suppose not. Then there 0 0 0 0 are a ∈ A, b ∈ B such that [a, a ) ∩ [b, b ) 6= ∅. Thus either b ∈ [a, a ) or a ∈ [b, b ), so either 0 0 0 0 [a, a ) ∩ B 6= ∅ or [b, b ) ∩ A 6= ∅, contrary to the choice of a and b .
Load more

Recommended publications
  • Nearly Metacompact Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 98 (1999) 191–201 Nearly metacompact spaces Elise Grabner a;∗, Gary Grabner a, Jerry E. Vaughan b a Department Mathematics, Slippery Rock University, Slippery Rock, PA 16057, USA b Department Mathematics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA Received 28 May 1997; received in revised form 30 December 1997 Abstract A space X is called nearly metacompact (meta-Lindelöf) provided that if U is an open cover of X then there is a dense set D ⊆ X and an open refinement V of U that is point-finite (point-countable) on D: We show that countably compact, nearly meta-Lindelöf T3-spaces are compact. That nearly metacompact radial spaces are meta-Lindelöf. Every space can be embedded as a closed subspace of a nearly metacompact space. We give an example of a countably compact, nearly meta-Lindelöf T2-space that is not compact and a nearly metacompact T2-space which is not irreducible. 1999 Elsevier Science B.V. All rights reserved. Keywords: Metacompact; Nearly metacompact; Irreducible; Countably compact; Radial AMS classification: Primary 54D20, Secondary 54A20; 54D30 A space X is called nearly metacompact (meta-Lindelöf) provided that if U is an open cover of X then there is a dense set D ⊆ X and an open refinement V of U such that Vx DfV 2 V: x 2 V g is finite (countable) for all x 2 D. The class of nearly metacompact spaces was introduced in [8] as a device for constructing a variety of interesting examples of non orthocompact spaces.
    [Show full text]
  • Or Dense in Itself)Ifx Has No Isolated Points
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 11, Pages 3607{3616 S 0002-9939(03)06660-7 Article electronically published on February 24, 2003 TYCHONOFF EXPANSIONS BY INDEPENDENT FAMILIES WANJUN HU (Communicated by Alan Dow) Abstract. A method for Tychonoff expansions using independent families is introduced. Using this method we prove that every countable Tychonoff space which admits a partition into infinitely many open-hereditarily irre- solvable dense subspaces has a Tychonoff expansion that is !-resolvable but not strongly extraresolvable. We also show that, under Luzin's Hypothesis ! ! (2 1 =2 ), there exists an !-resolvable Tychonoff space of size !1 which is not maximally resolvable. Introduction A topological space X is called crowded (or dense in itself)ifX has no isolated points. E. Hewitt defined a resolvable space as a space which has two disjoint dense subsets (see [17]). A space is called κ-resolvable [3] if it admits κ-many pairwise disjoint dense sets. The cardinal ∆(X):=minfjUj : U is a non-empty open subset of Xg is called the dispersion character of the space X.ThespaceX is called maximally resolvable [2] if it is ∆(X)-resolvable; if every crowded subspace of X is not resolvable, then X is called hereditarily irresolvable (HI). Analogously, X is open-hereditarily irresolvable (or OHI) if every non-empty open subset of X is irresolvable. V.I. Malykhin introduced a similar concept of extraresolvable (see [20] + and [6]): a space is extraresolvable if there exists a family fDα : α<∆(X) g of dense susbets of X such that Dα \ Dβ is nowhere dense in X for distinct α and β.
    [Show full text]
  • A Guide to Topology
    i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition.
    [Show full text]
  • 1.1.5 Hausdorff Spaces
    1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x).
    [Show full text]
  • The Non-Hausdorff Number of a Topological Space
    THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum.
    [Show full text]
  • On Dimension and Weight of a Local Contact Algebra
    Filomat 32:15 (2018), 5481–5500 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1815481D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Dimension and Weight of a Local Contact Algebra G. Dimova, E. Ivanova-Dimovaa, I. D ¨untschb aFaculty of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria bCollege of Maths. and Informatics, Fujian Normal University, Fuzhou, China. Permanent address: Dept. of Computer Science, Brock University, St. Catharines, Canada Abstract. As proved in [16], there exists a duality Λt between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then t t w(X) = wa(Λ (X)), and if, in addition, X is normal, then dim(X) = dima(Λ (X)). 1. Introduction According to Stone’s famous duality theorem [43], the Boolean algebra CO(X) of all clopen (= closed and open) subsets of a zero-dimensional compact Hausdorff space X carries the whole information about the space X, i.e. the space X can be reconstructed from CO(X), up to homeomorphism. It is natural to ask whether the Boolean algebra RC(X) of all regular closed subsets of a compact Hausdorff space X carries the full information about the space X (see Example 2.5 below for RC(X)).
    [Show full text]
  • 1. Introduction in a Topological Space, the Closure Is Characterized by the Limits of the Ultrafilters
    Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 08–37 THE ULTRAFILTER CLOSURE IN ZF GONC¸ALO GUTIERRES Abstract: It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with uX (A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈U]}. However, it is possible to built a topological space X for which uX 6= kX , but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {R}. Keywords: Ultrafilter Theorem, Ultrafilter Closure. AMS Subject Classification (2000): 03E25, 54A20. 1. Introduction In a topological space, the closure is characterized by the limits of the ultrafilters. Although, in the absence of the Axiom of Choice, this is not a fact anymore.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • Chapter 7 Separation Properties
    Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and .
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]
  • INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
    INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C.
    [Show full text]