LECTURE NOTES ON GENERAL TOPOLOGY BIT, SPRING 2021

DAVID G.L. WANG

Contents

1. Introduction 2 1.1. Who cares topology? 2 1.2. Geometry v.s. topology 2 1.3. The origin of topology 3 1.4. Topological equivalence 5 1.5. Surfaces 6 1.6. Abstract spaces 6 1.7. The classification theorem and more 6 2. Topological Spaces 8 2.1. Topological structures 8 2.2. Subspace topology 12 2.3. Point position with respect to a set 14 2.4. Bases of a topology 19 2.5. Metrics & the metric topology 21 3. Continuous Maps & Homeomorphisms 30 3.1. Continuous maps 30 3.2. Covers 35 3.3. Homeomorphisms 37 4. Connectedness 45 4.1. Connected spaces 45

Date: March 2, 2021. 1 4.2. Path-connectedness 53 5. Separation Axioms 57

5.1. Axioms T0, T1, T2, T3 and T4 57 5.2. Hausdor↵spaces 60 5.3. Regular spaces & normal spaces 62 5.4. Countability axioms 65 6. Compactness 68 6.1. Compact spaces 68 6.2. Interaction of compactness with other topological properties 70 7. Product Spaces & Quotient Spaces 75 7.1. Product spaces 75 7.2. Quotient spaces 81 Appendix A. Some elementary inequalities 88 3

1. Introduction

1.1. Who cares topology? The Nobel Prize in Physics 2016 was awarded with one half to David J. Thouless, and the other half to F. Duncan M. Haldane and J. Michael Kosterlitz “for theoretical discoveries of topological phase transitions and topo- logical phases of matter”; see Fig. 1.

Figure 1. Topology was the key to the Nobel Laureates’ discoveries, and it explains why electronical conductivity inside thin layers changes in integer steps. Stolen from Popular Science Background of the Nobel Prize in Physics 2016, Page 4(5).

Topology, over most of its history, has NOT generally been applied outside of mathematics (with a few interesting exceptions). WHY?

TOO abstract? The ancient mathematicians could not even convince of the subject. • It is qualitative, not quantitative? People think of science as a quantitative endeavour. •

1.2. Geometry v.s. topology. Below are some views from Robert MacPherson, a plenary addresser at the ICM in Warsaw in 1983.

Geometry (from ancient Greek): geo=earth, metry=measurement. • Topology (from Greek): ⌧´o ⇡o=place/position, ´o o=study/discourse. 4 D.G.L. WANG

Figure 2. Screenshot from a video of Robert MacPherson’s talk in Institute for Advanced Study.

Topology is “geometry” without measurement. • It is qualitative (as opposed to quantitative) “geometry”. Geometry: The point M is the midpoint of the straight line segment L connecting • A to B. Topology: The point M lies on the curve L connecting A to B. Geometry calls its objects configurations (circles, triangles, etc.) • Topology calls its objects spaces.

1.3. The origin of topology. Here are three stories about the origin of topology.

1.3.1. The seven bridges of K¨onigsberg. The problem was to devise a walk through the city that would cross each of those bridges once and only once; see Fig. 3. The negative resolution by Leonhard Paul Euler (1707–1783) in 1736 laid the founda- tions of graph theory and prefigured the idea of topology. The diculty Euler faced was the development of a suitable technique of analysis, and of subsequent tests that established this assertion with mathematical rigor. Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics like infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

1.3.2. The four colour theorem. The four colour theorem states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the map so that no two adjacent regions have the same color. 5

Figure 3. Map of K¨onigsberg in Euler’s time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. Stolen from Wiki.

The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.

1.3.3. Euler characteristic. = v e + f; see Fig. 4.

Figure 4. Stamp of the former German Democratic Republic honouring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula. Stolen from Wiki.

We use the terminology polyhedron to indicate a surface rather than a solid. Theorem 1.1 (Euler’s polyhedral formula). Let P be a polyhedron s.t.

Any two vertices of P can be connected by a chain of edges. • Any cycle along edges of P which is made up of straight line segments (not necessarily • edges) separates P into 2 pieces.

Then the Euler number or Euler characteristic =2for P .

1750: first appear in a letter from Euler to Christian Goldbach (1690–1764). • 1860 (around): M¨obiusgave the idea of explaining topological equivalence by thinking • of spaces as being made of rubber.ItworksforconcaveP . 6 D.G.L. WANG

David Eppstein collected 20 proofs of Theorem 1.1. • In 3-dimensional space, a Platonic solid is a regular, convex polyhedron. The Euler characteristic of every Platonic solid is 2. In fact, there are only 5 (why?) Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron; see Fig. 5.

Figure 5. The Plotonic solids. Stolen from Wiki.

1.4. Topological equivalence. =homeomorphism;seeSection1.6.

p :thicken,stretch,bend,twist,...; :identify,tear,.... ⇥ See Fig. 6.

Figure 6. Some surfaces which are not equivalent. Stolen from Haldane’s slides on Dec. 8th, 2016. 7

Theorem 1.2. Topological equivalent polyhedra have the same Euler characteristic.

The starting point for modern topology. • Search for unchanged properties of spaces under topological equivalence. • =2belongstoS2,ratherthantoparticularpolyhedra define for S2. • ! Theorem 1.2:di↵erentcalculations,sameanswer. • 1.5. Surfaces. What exactly do we mean by a “space”?

Homeomorphism Continuity. • ! Geometry Boundedness. • ! 1.6. Abstract spaces. The axioms for a topological space appearing for the first time in 1914 in the work of Felix Hausdor↵ (1868–1942). Hausdor↵, a German mathematician, is considered to be one of the founders of modern topology, who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis. How has modern definition of a topological space been formed?

(1) General enough to allow set of points or functions, and performable constructions like the Cartesian products and the identifying. Enough information to define the continuity of functions between spaces. (2) Cauchy: distances continuity. ! (3) No distance! Continuity neighbourhood axiom.

Afunctionf : Em En is continuous if given any x Em and any neighbourhood 1 ! 2 U of f(x), then f (U)isaneighbourhoodofx.

1.7. The classification theorem and more. Theorem 1.3 (Classification theorem). Any closed surface is homeomorphic to S2 with either a finite number of handles added, or a finite number of M¨obiusstrips added. No two of these surfaces are homeomorphic. Definition 1.4. The S2 with n handles added is called an orientable surface of genus n. Non-orientable surfaces can be defined analogously.

Historical notes. The classification of surfaces was initiated and carried through in the orientable case by M¨obiusin a paper which he submitted for consideration for the Grand Prix de Math´ematiques of the Paris Academy of Sciences. He was 71 at the time. The jury did not consider any of the manuscripts received as being worthy of the prize, and M¨obius’ work finally appeared as just another mathematical paper. 8 D.G.L. WANG

Decide = or =. ⇠ 6⇠ =: construct a homeomorphism; techniques vary. • ⇠ ⇠=:lookfortopologicalinvariants,e.g.,geometricproperties,numbers,algebraic • systems.6

Examples to show =. 6⇠ E1 = E2:connectedness,h: E1 0 E2 h(0) . • 6⇠ \{ }! \{ } Poincar´e’s construction idea: assign a group to each topological space so that • homeomorphic spaces have isomorphic groups. But group isomorphism does not imply homeomorphism.

Here are some theorems that the fundamental groups help prove. Theorem 1.5 (Classification of surfaces). No 2 surfaces in Theorem 1.3 have isomorphic fundamental groups. Theorem 1.6 (Jordan separation theorem). Any simple closed curve in E1 divides E1 into 2 pieces.

See Fig. 7.

Figure 7. Marie Ennemond Camille Jordan (1838–1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d’analyse. The Jordan curve (drawn in black) divides the plane into an “inside” region (light blue) and an “outside” region (pink). Stolen from Wiki.

Theorem 1.7 (Brouwer fixed-point theorem). Any continuous function from a disc to itself leaves at least one point fixed.

See Fig. 8. Theorem 1.8 (Nielsen-Schreier theorem). A subgroup of a free group is always free. 9

Figure 8. Luitzen Egbertus Jan Brouwer (1881–1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathe- matician and philosopher, who worked in topology, set theory, measure theory and complex analysis. He was the founder of the mathematical philosophy of intuitionism. Stolen from Wiki.

2. Topological Spaces

The definition of topological space fits quite well with our intuitive idea of what a space ought to be. Unfortunately it is not terribly convenient to work with. We want an equivalent, more manageable, set of axioms!

2.1. Topological structures. Definition 2.1. Let X be a set and ⌦ 2X . ✓ A(topological)space is a pair (X, ⌦), where the collection ⌦, called a topology or • topological structure on X,satisfiestheaxioms (i) ⌦andX ⌦; ;2 2 (ii) the union of any members of ⌦lies in ⌦; (iii) the intersection of any two members of ⌦lies in ⌦. A point in X: p X. • 2 An open set in (X, ⌦): a member in ⌦. • A closed set in (X, ⌦): a subset A X s.t. Ac ⌦. ✓ 2 A clopen set in (X, ⌦): a subset A X which is both closed and open. ✓ Remark 2.2. Being closed is not the negation of being open! A set might be open but not closed, or • closed but not open, or • clopen, or • neither closed nor open. • Remark 2.3. Why do we use the letter O and the letter ⌦? Open in English • 10 D.G.L. WANG

Ouvert in French • Otkrytyj in Russian • O↵en in German • Oppen¨ in Swedish • Otvoren in Croatian • Otevˇreno in Czech • Open in Dutch •

Here are some topological spaces that we will meet frequently in this note. Space 1. An indiscrete or trivial topological space: (X, ,X ). {; } Space 2. A discrete topological space: (X, 2X ). Aspaceisdiscrete every singleton is open. () Space 3. A particular point topology (X, ⌦): ⌦= ,X S X : p S , {; }[{ ✓ 2 } where p is a particular point in X.

Space 4. An excluded point topology (X, ⌦): ⌦= ,X S X : p S , {; }[{ ✓ 62 } where p is a particular point in X.

Space 5. The real line (R, ⌦R): ⌦ = unions of open intervals R { } is the canonical or standard topology on R.

Space 6. The cofinite space (R, ⌦T1 ):

⌦T1 = complements of finite subsets of R {;}[{ } is the cofinite topology or finite-complement topology or T1-topology.

Space 7. The arrow (X, ⌦): X = x R: x 0 and ⌦= ,X (a, ): a 0 . { 2 } {; }[{ 1 } Space 8. The Sierpi´nski space (X, ⌦): X = a, b and ⌦= , a , a, b . { } {; { } { }} 11

Figure 9. Waclaw Franciszek Sierpi´nski (1882–1969) was a Polish math- ematician. He was known for outstanding contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions and topology. He published over 700 papers and 50 books. Stolen from Wiki.

Homework 2.1. Find a smallest topological space which is neither discrete nor indiscrete.

Question 2.4. Does there exist a topology which is both a particular point topology and an excluded point topology?

Example 2.5. The set 0 1/n: n Z+ is closed in the real line. { }[{ 2 } ⇤ “Think geometrically, prove algebraically.” — John Tate

Figure 10. John Tate (1925–) is an American mathematician, distin- guished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is pro- fessor emeritus at Harvard Univ. He was awarded the Abel Prize in 2010. Stolen from Wiki.

Homework 2.2. Find a topological space (X, ⌦) with a set A X satisfying all the following properties: ⇢ 12 D.G.L. WANG a) A is neither open nor closed; b) A is the union of an infinite number of closed sets; and c) A is the intersection of an infinite number of open sets.

Answer. The interval [ 0, 1) in R. ⇤

Example 2.6. The Cantor ternary set K is the number set created by iteratively deleting the open middle third from a set of line segments, i.e., a K = k : a 0, 2 = 0.a a : a 1, 4, 7 [0, 1]. 3k k 2{ } 1 2 ··· i 62 { } ⇢ ⇢ k 1 X n o See Fig. 11.ThesetK was discovered by Henry John Stephen Smith in 1874, and

Figure 11. The left most is Henry John Stephen Smith (1826–1883), a mathematician remembered for his work in elementary divisors, quadratic forms, and Smith-Minkowski-Siegel mass formula in number theory. The middle is Georg Cantor (1845–1918), a German mathematician who invented set theory. The right most illustrates the Cantor ternary set. Stolen from Wiki and Math Counterexamples respectively.

introduced by Georg Cantor in 1883. It has a number of remarkable and deep properties. For instance, it is closed in R.

Definition 2.7. Given (X, ⌦). Let p X.Aneighbourhood of p is a subset U X s.t. 2 ✓ O ⌦s.t.p O U. 9 2 2 ✓ Remark 2.8. In literature the letter U is used to indicate a neighbourhood since it is the first letter of the German word “umgebung” which means neighbourhood. Remark 2.9. We are following the Nicolas Bourbaki group and define the term “neigh- bourhood” in the above sense. There is another custom that a neighbourhood of a point p is an open set containing p. Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim 13

of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics. While there is no one person named Nicolas Bourbaki, the Bourbaki group, ocially known as the Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), has an oce at the Ecole´ Normale Sup´erieure in Paris.

Question 2.10. Atopologycanbedefinedbyassigningneighbourhoodsoropensets; see Definition 2.1.Canitbedefinedbyassigningclosedsets?

Answer. Yes. Here is a list of axioms for assigning closed sets:

(i)’ and X are closed; ; (ii)’ the union of any finite number of closed sets is closed; (iii)’ the intersection of any collection of closed sets is closed.

⇤ Remark 2.11. Given (X, ⌦). Since the union of all members in ⌦is X,thetopology⌦ itself carries enough information to clarify a topological space. However, the topological space (X, ⌦) is often denoted simply by X,becausedi↵erenttopologicalstructuresin the same set X are often considered simultaneously rather seldom. Moreover, to exclaim a set is in general easier than clarifying a topology. As will be seen in Section 2.2, subspace topology helps the clarification by calling the knowledge of common topologies.

2.2. Subspace topology. Definition 2.12. Given (X, ⌦) and A X.Thesubspace topology (or induced topology)ofA induced from (X, ⌦): ✓ ⌦ = O A: O ⌦ . A { \ 2 } The topological subspace induced by A:(A, ⌦A).

Question 2.13. Describe the topological structures induced

+ 1) on Z by ⌦R;

Answer. The discrete topology. ⇤

2) on Z+ by the arrow;

Answer. All sets of the form n N: n a where a N. { 2 } 2 ⇤ 14 D.G.L. WANG

3) on the two-element set 1, 2 by ⌦ ; { } T1 Answer. The discrete topology. ⇤ 4) on the two-element set 1, 2 by the arrow topology. { } Answer. , 2 , 1, 2 . {; { } { }} ⇤ Theorem 2.14. Let (X, ⌦) be a topological space and A X. The subspace topology of A induced from (X, ⌦) can be defined alternatively in terms✓ of closed sets as S is closed in A S = F A, where F is a closed set in X. () \ ⇤ Theorem 2.15. Let X be a topological space and let A M X. ✓ ✓ 1) If A is open in X, then A is open in M. If A is closed in X, then A is closed in M. 2) If A is open in M, and if M is open in X, then A is open in X. If A is closed in M, and if M is closed in X, then A is closed in X.

Proof. The openness of A in M follows from the formula A = A M. Conversely, if A is open in M,thenA = O M,whereO is open in X. As\ a consequence, this intersection A = O M is open\ in X as long as M is also open in X.The“closed” \ version can be shown along the same line with aid of Theorem 2.14. ⇤ Remark 2.16. The condition that M is open/closed in X in Theorem 2.15 is necessary. For instance, the set x Q: x>p2 is clopen in Q,butneitherclosednoropeninR. { 2 } Homework 2.3. Let A M X. ✓ ✓ (1) If A is open in M,canweinferthatA is open in X? (2) If A is closed in M,canweinferthatA is closed in X?

Answer. No to both. ⇤

Theorem 2.15 is about the openness of A in the set inclusion structure A M X. Theorem 2.17 concerns the topology of A in that structure. ✓ ✓ Theorem 2.17. Let A M X. Then the topology of A induced from the topology ✓ ✓ of X coincides with the topology of A induced from ⌦M , where ⌦M is the subspace topology of M induced from the topology of X. In other words, it is safe to say “the subspace topology of A”.

Proof. O A: O ⌦ = (O0 M) A: O0 ⌦ = O0 A: O0 ⌦ . { \ 2 M } { \ \ 2 X } { \ 2 X } ⇤ 15

2.3. Point position with respect to a set. Definition 2.18. Given (X, ⌦) and A X. ✓ A limit point of A:apointp X s.t. • 2 U (A p ) = , neighbourhood U of p. \ \{ } 6 ; 8 An isolated point of A:apointp A s.t. 2 aneighbourhoodU of p s.t. U (A p )= . 9 \ \{ } ; A is perfect,ifitisclosedandhasnoisolatedpoints. • The closure of A:theunionofA and its limit points, denoted A,i.e., • A = x X : N A = , neighbourhood N of x . { 2 \ 6 ; 8 } It is alternatively written as ClA when considered as a set operator. An adherent point of A:apointinA. An interior point of A:apointhavinganeighbourhoodinA. • An exterior point of A: a point having a neighbourhood in Ac. A boundary point of A:apoints.t.eachneighbourhoodmeetsbothA and Ac.

The interior of A: A = interior points = O ⌦: O A ; • { } [{ 2 ✓ } c The exterior of A w.r.t. X:(A ) = exterior points ; { } The boundary of A: @A = boundary points = A A. { } \ Remark 2.19. The symbols Cl(A), Int(A), and Ext(A)areusedtodenotetheclosureA, c the interior A,andtheexterior(A ) resp., when one emphasizes that they are set operators. The symbol Bd(A)issometimesrecognizedastheboundaryofA,butused uncommonly since the symbol @ well plays the role of a set operator.

Example 2.20. A X,wehave 8 ✓ c A A, ExtA A , and Ext = X. ✓ ✓ ; In R,wehaveIntQ = and ExtQ = .FortheCantorsetinExample2.6,wehave ; ; K = K, K = , ExtK = I K, and @K = K. ; \ Notation 2.21. The unit interval:theinterval[0, 1] in R,denotedbyI.

Puzzle 2.22. The Smith-Volterra-Cantor set K0 is a set of points on the real line R; see Fig. 12. It can be obtained by removing certain intervals from I as follows. After removing the middle 1/4fromI,removethesubintervalsoflength 1/4n from the middle of each of the remaining intervals. For instance, at the first and second step the remaining intervals are 3 5 5 7 3 5 25 27 0, , 1 and 0, , , , 1 . 8 [ 8 32 [ 32 8 [ 8 32 [ 32       Show that the set K0 is nowhere dense. 16 D.G.L. WANG

Figure 12. The Smith-Volterra-Cantor set. Stolen from Bing image.

Theorem 2.23 (Characterization of the closure and interior). The closure of a set is the smallest closed set containing that set, and the interior of a set is the largest open set contained in that set.

Proof. The closure of a set is the intersection of all closed sets containing it, and the interior of a set is the union of all open sets contained in it. ⇤

Proposition 2.24. Given a topological space X.

1) The interior and exterior of a set are open, and the boundary is closed. 2) Any set A in X can be decomposed as A =Lim(A) Iso(A), t where Lim(A) and Iso(A) are the sets of limits and isolated points of A resp. 3) The whole set X can be decomposed w.r.t. a subset A: X =(@A) (IntA) (ExtA)=(ClA) (ExtA), t t t 4) @A = A Ac. \ 5) @(A) @A. ✓

Definition 2.25. Let f be an operator.

f is an involution: f f =id. is the identity map. • f is idempotent: f f = f. • In the operator theory, we denote the set operator of closure, interior, complement resp. by k, i,andc.Forinstance,thesetoperatorc is an involution.

Theorem 2.26. Given (X, ⌦) and A X. ✓ 1) Both the set operations interior and closure preserve the inclusion, i.e.,

A X = A X and A X. ✓ ) ✓ ✓ 2) ci = kc. 3) The operators k, i, and ki = kckc are idempotent. 17

4) The distributivity of k with , and the distributivity of i with : [ \ A B = A B and (A B) = A B. [ [ \ \ The operators k does not work well with , neither does i with : \ [ A B A B and (A B) A B. \ ✓ \ [ ◆ [ Remark 2.27. It is clear that the exterior does not preserves the inclusion, and that it is not idempotent. From Theorem 2.26,onemayseethatthesetoperationclosureis “more dual” to the set operation interior, comparing to the set operator exterior.

However, there are some properties that closure and interior do not share. Theorem 2.28. Let X be a topological space and A M X. Then ✓ ✓ Cl (A)=Cl (A) M, M X \ but Int (A) =Int (A) M in general. M 6 X \ Proof. The formula for closure holds since

ClM (A)= F = (C M)=M C = M ClX (A), A F A C \ \ A C \ F is closed\✓ in M C is closed\✓ in X C is closed\✓ in X The other formula holds false for instance X = R2, M = A = R. ⇤

Puzzle 2.29 (Kuratowski’s closure-complement problem). How many pairwise distinct sets can one obtain from of a given subset of a topological space by using the set operators k and c?

Figure 13. Kazimierz Kuratowski (1896–1980) was a Polish mathemati- cian and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Stolen from Wiki.

Answer. 14. An example in the real line: (0, 1) (1, 2) 3 ([4, 5] Q). [ [{ }[ \ ⇤ 18 D.G.L. WANG

The answer 14 was first published by Kuratowski in 1922. A subset realising the maximum of 14 is called a 14-set.

Puzzle 2.30. Recall that we can define a topology either in terms of open sets, or in terms of closed sets, or in terms of neighbourhoods. Can we define a topology with the aid of the closure operation, or the interior operation?

Answer. Let X be a set. Let Cl be a transformation on the power set 2X s.t. ⇤ (i) Cl = ; ⇤; ; (ii) A Cl A; ✓ ⇤ (iii) distributive with the union operation: Cl (A B)=ClA Cl B; ⇤ [ ⇤ [ ⇤ (iv) idempotent: Cl Cl A =ClA. ⇤ ⇤ ⇤ Then the set ⌦= O X :Cl(Oc)=Oc is a topology on X.Moreover,thesetCl A { ✓ ⇤ } ⇤ is the closure of a set A in the topological space (X, ⌦). ⇤

Definition 2.31. Given (X, ⌦) and A, B X. ✓ A is dense in B: B A. • ✓ A is everywhere dense: A = X,i.e.,ExtA = . • ; A is nowhere dense:ExtA is everywhere dense, i.e., ExtExtA = . • ; Remark 2.32. Concerning topics on subset density, an often helpful fact is Theorem 2.23; see the proofs of Theorem 2.34 and ?? 2.35?? 2.36.

Example 2.33. The whole set is everywhere dense, and the empty set is not everywhere dense. Continuing Example 2.20,thesetQ is everywhere dense in R,andtheCantor set is nowhere dense in I.

Theorem 2.34 (Characterization for everywhere density). A set is everywhere dense it meets every nonempty open set it meets every neighbourhood. () () Proof. The second equivalent is clear. We show the first. Let (X, ⌦) be a topological space with A X. ✓ . Let O ⌦s.t.A O = .ThenA is a subset of the closed set Oc.Itfollowsthat ) X = A2 Oc. Hence\ O =; . ✓ ; . Let F X be a proper closed set containing A. From premise, we have A F c = ( ,thatis,⇢ A F .Therefore,thesmallestclosedsetcontainingA is X\.By6 Theorem; 2.23,wehave6✓ A = X. Hence A is everywhere dense. 19

This completes the proof. ⇤

Corollary 2.35. If A is everywhere dense and O is open, then O A O. ✓ \ Proof. If not, then x O with a neighbourhood U s.t. U (A O)= . W.l.o.g., we can suppose that9 U 2is open. Then the everywhere dense\ set A\does not; meet the open set U O.ByTheorem2.34,weinferthatU O = ,contradictingthefactthat x U O.\ \ ; 2 \ ⇤ Corollary 2.36. Let X be a topological space.

(1) X is indiscrete only the empty set is not everywhere dense. () ; Proof. Let A X s.t. A ,X .IfX is indiscrete, then there is only one nonempty open set, that✓ is, the whole62 {; set X}.CertainlyA meets X.ByTheorem2.34,theset A is everywhere dense. Conversely, since A = X = A,noA is closed. Hence X is 6 indiscrete. ⇤ (2) X is discrete only the whole set X is everywhere dense. () Proof. Let A X s.t. A ,X .IfX is discrete, then A is closed, and A = A = X. Conversely, by✓ Theorem622.34 {; ,theset} A does not meet some nonempty open6 set. Taking A to be the complement of each singleton, we find that every singleton is open. Hence X is discrete. ⇤ (3) A set S is everywhere dense in the arrow sup S = . () 1 Proof. If sup S = ,then(s, ) S = , s 0. It is clear that [ 0, ) S = . Thus S meets every1 nonempty1 open\ set,6 ; and8 is everywhere dense by Theorem1 \ 2.346 ;. Conversely, if sup S = ,then M>0s.t.s

Proof. By Theorem 2.34,wededucethatA is nowhere dense ExtA is everywhere dense () ExtA meets every neighbourhood () each neighbourhood contains an exterior point of A () each neighbourhood contains a neighbourhood that is contained in Ac () each open set contains an open set that is contained in Ac. () This completes the proof. ⇤