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 onehalf to David J. Thouless, and the other half to F. Duncan M. Haldane and J. MichaelKosterlitz “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 changesin integer steps. Stolen from Popular Science Background of the NobelPrize in Physics 2016, Page 4(5).

Topology, over most of its history, has NOT generally been applied outside ofmathematics (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, aplenary 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 inInstitute 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 connectingA 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 Konigsberg. The problem was to devise a walk through thecity 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 di�culty Euler facedwas the development of a suitable technique of analysis, and of subsequent tests thatestablished this assertion with mathematical rigor.

Euler was a Swiss mathematician, physicist, astronomer, logician and engineer whomade important and influential discoveries in many branches of mathematics likeinfinitesimal calculus and graph theory, while also making pioneering contributions toseveral branches such as topology and analytic number theory.

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

5

Figure 3. Map of Konigsberg in Euler’s time showing the actual layoutof the seven bridges, highlighting the river Pregel and the bridges. Stolenfrom 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 honouringEuler on the 200th anniversary of his death. Across the centre it showshis 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 necessarilyedges) separates P into 2 pieces.

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

• 1750: first appear in a letter from Euler to Christian Goldbach (1690–1764).

• 1860 (around): Mobius gave the idea of explaining topological equivalence by thinkingof spaces as being made of rubber. It works for concave P .

6 D.G.L. WANG

• David Eppstein collected 1.120 proofs of Theorem 1.1.

In 3-dimensional space, a Platonic solid is a regular, convex polyhedron. The Eulercharacteristic 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; seeFig. 5.

Figure 5. The Plotonic solids. Stolen from Wiki.

1.4. Topological equivalence. = homeomorphism; see Section 1.6.

p: thicken, stretch, bend, twist, . . .;

⇥ : identify, tear, . . ..

See Fig. 6.

Figure 6. Some surfaces which are not equivalent. Stolen from Haldane’sslides 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.

• � = 2 belongs to S2, rather than to particular polyhedra ! define � for S2.

• Theorem 1.2: di↵erent calculations, same answer.

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 in1914 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 significantlyto set theory, descriptive set theory, measure theory, function theory, and functionalanalysis.

How has modern definition of a topological space been formed?

(1) General enough to allow set of points or functions, and performable constructionslike the Cartesian products and the identifying. Enough information to define thecontinuity of functions between spaces.

(2) Cauchy: distances ! continuity.

(3) No distance! Continuity neighbourhood axiom.

A function f : Em! En is continuous if given any x 2 Em and any neighbourhood

U of f(x), then f�1(U) is a neighbourhood of x.

1.7. The classification theorem and more.

Theorem 1.3 (Classification theorem). Any closed surface is homeomorphic to S2 witheither a finite number of handles added, or a finite number of Mobius strips added. Notwo of these surfaces are homeomorphic.

Definition 1.4. The S2 with n handles added is called an orientable surface of genusn. Non-orientable surfaces can be defined analogously.

Historical notes. The classification of surfaces was initiated and carried through inthe orientable case by Mobius in a paper which he submitted for consideration for theGrand Prix de Mathematiques of the Paris Academy of Sciences. He was 71 at thetime. The jury did not consider any of the manuscripts received as being worthy of theprize, and Mobius’ work finally appeared as just another mathematical paper.

8 D.G.L. WANG

Decide ⇠= or 6⇠=.

• ⇠=: construct a homeomorphism; techniques vary.

• 6⇠=: look for topological invariants, e.g., geometric properties, numbers, algebraicsystems.

Examples to show 6⇠=.

• E16⇠= E2: connectedness, h : E1

\ {0}! E2\ {h(0)}.

• Poincare’s construction idea: assign a group to each topological space so thathomeomorphic spaces have isomorphic groups. But group isomorphism does notimply 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 isomorphicfundamental 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 Frenchmathematician, known both for his foundational work in group theoryand 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 toitself 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 asL. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathe-matician and philosopher, who worked in topology, set theory, measuretheory and complex analysis. He was the founder of the mathematicalphilosophy of intuitionism. Stolen from Wiki.

2. Topological Spaces

The definition of topological space fits quite well with our intuitive idea of what aspace ought to be. Unfortunately it is not terribly convenient to work with. We wantan 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 ortopological structure on X, satisfies the axioms

(i) ; 2 ⌦ and X 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 2 X.

• An open set in (X,⌦): a member in ⌦.

A closed set in (X,⌦): a subset A ✓ X s.t. Ac2 ⌦.

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

• Otevreno 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).

A space is discrete () every singleton is open.

Space 3. A particular point topology (X,⌦):

⌦ = {;, X} [ {S ✓ X : p 2 S},

where p is a particular point in X.

Space 4. An excluded point topology (X,⌦):

⌦ = {;, X} [ {S ✓ X : p 62 S},

where p is a particular point in X.

Space 5. The real line (R, ⌦R):

⌦R = {unions of open intervals}

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 2 R : x � 0} and ⌦ = {;, X} [ {(a,1) : a � 0}.

Space 8. The Sierpinski space (X,⌦):

X = {a, b} and ⌦ = { ;, {a}, {a, b}}.

11

Figure 9. Wac law Franciszek Sierpinski (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), numbertheory, theory of functions and topology. He published over 700 papersand 50 books. Stolen from Wiki.

Homework 2.1. Find a smallest topological space which is neither discrete norindiscrete.

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

Example 2.5. The set {0} [ {1/n : n 2 Z+} is closed in the real line. ⇤

“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 thefollowing 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 iterativelydeleting the open middle third from a set of line segments, i.e.,

K =

⇢X

k�1

ak3k

: ak 2 {0, 2}

�=

n0.a1a2 · · · : ai 62 {1, 4, 7}

o⇢ [0, 1].

See Fig. 11. The set K was discovered by Henry John Stephen Smith in 1874, and

Figure 11. The left most is Henry John Stephen Smith (1826–1883), amathematician remembered for his work in elementary divisors, quadraticforms, and Smith-Minkowski-Siegel mass formula in number theory. Themiddle is Georg Cantor (1845–1918), a German mathematician whoinvented 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 2 X. A neighbourhood of p is a subset U ✓ Xs.t.

9 O2 ⌦ s.t. p 2 O ✓ U.

Remark 2.8. In literature the letter U is used to indicate a neighbourhood since it isthe 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 apoint p is an open set containing p. Nicolas Bourbaki is the collective pseudonymunder which a group of (mainly French) 20th-century mathematicians, with the aim

13

of reformulating mathematics on an extremely abstract and formal but self-containedbasis, wrote a series of books beginning in 1935. With the goal of grounding all ofmathematics on set theory, the group strove for rigour and generality. Their workled to the discovery of several concepts and terminologies still used, and influencedmodern branches of mathematics. While there is no one person named Nicolas Bourbaki,the Bourbaki group, o�cially known as the Association des collaborateurs de NicolasBourbaki (Association of Collaborators of Nicolas Bourbaki), has an o�ce at the EcoleNormale Superieure in Paris.

Question 2.10. A topology can be defined by assigning neighbourhoods or open sets;see Definition 2.1. Can it be defined by assigning closed sets?

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, the topology ⌦itself carries enough information to clarify a topological space. However, the topologicalspace (X,⌦) is often denoted simply by X, because di↵erent topological structures inthe same set X are often considered simultaneously rather seldom. Moreover, to exclaima 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. The subspace topology (or inducedtopology) of A induced from (X,⌦):

⌦A = {O \ A : O 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 2 N : n � a} where a 2 N. ⇤

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 topologyof 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, ifA is open in M , then A = O \M , where O is open in X. As a consequence, thisintersection 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 2 Q : x >

p2} is clopen in Q, but neither closed nor open in R.

Homework 2.3. Let A ✓M ✓ X.

(1) If A is open in M , can we infer that A is open in X?

(2) If A is closed in M , can we infer that A 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 topologyof X coincides with the topology of A induced from ⌦M , where ⌦M is the subspacetopology of M induced from the topology of X. In other words, it is safe to say “thesubspace topology of A”.

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

2 ⌦X} = {O0\ A : O0

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: a point p 2 X s.t.

U \ (A \ {p}) 6= ;, 8 neighbourhood U of p.

An isolated point of A: a point p 2 A s.t.

9 a neighbourhood U of p s.t. U \ (A \ {p}) = ;.

• A is perfect, if it is closed and has no isolated points.

• The closure of A: the union of A and its limit points, denoted A, i.e.,

A = {x 2 X : N \ A 6= ;, 8 neighbourhood N of x}.

It is alternatively written as ClA when considered as a set operator.

An adherent point of A: a point in A.

• An interior point of A: a point having a neighbourhood in A.

An exterior point of A: a point having a neighbourhood in Ac.

A boundary point of A: a point s.t. each neighbourhood meets both A and Ac.

• The interior of A: A� = {interior points} = [{O 2 ⌦ : O ✓ A};

The exterior of A w.r.t. X: (Ac)� = {exterior points};

The boundary of A: @A = {boundary points} = A \ A�.

Remark 2.19. The symbols Cl(A), Int(A), and Ext(A) are used to denote the closure A,the interior A�, and the exterior (Ac)� resp., when one emphasizes that they are setoperators. The symbol Bd(A) is sometimes recognized as the boundary of A, but useduncommonly since the symbol @ well plays the role of a set operator.

Example 2.20. 8A ✓ X, we have

A�✓ A, ExtA ✓ Ac, and Ext; = X.

In R, we have IntQ = ; and ExtQ = ;. For the Cantor set in Example 2.6, we have

K = K, K� = ;, ExtK = I \K, and @K = K.

Notation 2.21. The unit interval: the interval [0, 1] in R, denoted by I.

Puzzle 2.22. The Smith-Volterra-Cantor set K 0 is a set of points on the real lineR; see Fig. 12. It can be obtained by removing certain intervals from I as follows. Afterremoving the middle 1/4 from I, remove the subintervals of length 1/4n from themiddle of each of the remaining intervals. For instance, at the first and second step theremaining intervals are

0,

3

8

�[

5

8, 1

�and

0,

5

32

�[

7

32,3

8

�[

5

8,25

32

�[

27

32, 1

�.

Show that the set K 0 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 isthe smallest closed set containing that set, and the interior of a set is the largest openset contained in that set.

Proof. The closure of a set is the intersection of all closed sets containing it, and theinterior 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) t Iso(A),

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) t (IntA) t (ExtA) = (ClA) t (ExtA),

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, complementresp. by k, i, and c. For instance, the set operator c 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 itis not idempotent. From Theorem 2.26, one may see that the set operation closure is“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

ClM(A) = ClX(A) \M,

but IntM(A) 6= IntX(A) \M in general.

Proof. The formula for closure holds since

ClM(A) =\

A✓FF is closed in M

F =\

A✓CC is closed in X

(C \M) = M \\

A✓CC is closed in X

C = M \ ClX(A),

The other formula holds false for instance X = R2, M = A = R. ⇤

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

Figure 13. Kazimierz Kuratowski (1896–1980) was a Polish mathemati-cian and logician. He was one of the leading representatives of the WarsawSchool 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 themaximum of 14 is called a 14-set.

Puzzle 2.30. Recall that we can define a topology either in terms of open sets, or interms of closed sets, or in terms of neighbourhoods. Can we define a topology with theaid 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) = Cl⇤A [ Cl⇤B;

(iv) idempotent: Cl⇤Cl⇤A = Cl⇤A.

Then the set ⌦ = {O ✓ X : Cl⇤(Oc) = Oc} is a topology on X. Moreover, the set Cl⇤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 everywheredense. Continuing Example 2.20, the set Q is everywhere dense in R, and the Cantorset 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 topologicalspace with A ✓ X.

). Let O 2 ⌦ s.t. A \O = ;. Then A is a subset of the closed set Oc. It follows thatX = A ✓ Oc. Hence O = ;.

(. Let F ⇢ X be a proper closed set containing A. From premise, we have A \ F c6=

;, that is, A 6✓ F . Therefore, the smallest closed set containing A is X. ByTheorem 2.23, we have 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 9 x 2 O with a neighbourhood U s.t. U \ (A \ O) = ;. W.l.o.g.,we can suppose that U is open. Then the everywhere dense set A does not meet theopen set U \O. By Theorem 2.34, we infer that U \O = ;, contradicting the fact thatx 2 U \O. ⇤

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 62 {;, X}. IfX is indiscrete, then there is only one nonemptyopen set, that is, the whole set X. Certainly A meets X. By Theorem 2.34, the setA is everywhere dense. Conversely, since A = X 6= A, no A is closed. Hence X isindiscrete. ⇤

(2) X is discrete () only the whole set X is everywhere dense.

Proof. Let A ✓ X s.t. A 62 {;, X}. If X is discrete, then A is closed, and A = A 6= X.Conversely, by Theorem 2.34, the set A does not meet some nonempty open set.Taking A to be the complement of each singleton, we find that every singleton isopen. Hence X is discrete. ⇤

(3) A set S is everywhere dense in the arrow () supS =1.

Proof. If supS =1, then (s,1) \ S 6= ;, 8 s � 0. It is clear that [ 0,1) \ S 6= ;.Thus S meets every nonempty open set, and is everywhere dense by Theorem 2.34.Conversely, if supS 6= 1, then 9 M > 0 s.t. s < M , 8 s 2 S. It follows thatS \ (M,1) = ;. By Theorem 2.34, we infer that S is not everywhere dense. ⇤

(4) A set S is everywhere dense in the cofinite space () S is infinite. ⇤

Theorem 2.37. A set A is nowhere dense in a topological space X ()

each open set in X contains an open set that is contained in Ac.

Proof. By Theorem 2.34, we deduce that A 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. ⇤