Topology Course Lecture Notes

Aisling McCluskey and Brian McMaster

August 1997

Chapter 1
Fundamental Concepts

In the study of metric spaces, we observed that:

(i) many of the concepts can be described purely in terms of open sets,
(ii) open-set descriptions are sometimes simpler than metric descriptions, e.g. continuity,
(iii) many results about these concepts can be proved using only the basic properties of open sets (namely, that both the empty set and the underlying set X are open, that the intersection of any two open sets is again open and that the union of arbitrarily many open sets is open).

This prompts the question: How far would we get if we started with a collection of subsets possessing these above-mentioned properties and proceeded to define everything in terms of them?

1.1  Describing Topological Spaces

We noted above that many important results in metric spaces can be proved using only the basic properties of open sets that

• the empty set and underlying set X are both open,
• the intersection of any two open sets is open, and
• unions of arbitrarily many open sets are open.

We will call any collection of sets on X satisfying these properties a topology. In the following section, we also seek to give alternative ways of describing this important collection of sets.

1.1.1  Defining Topological Spaces

Definition 1 A topological space is a pair (X,T) consisting of a set X and a family T of subsets of X satisfying the following conditions:

(T1) Æ Î T and X Î T
(T2) T is closed under arbitrary union
(T3) T is closed under finite intersection.

The set X is called a space, the elements of X are called points of the space and the subsets of X belonging to T are called open in the space; the family T of open subsets of X is also called a topology for X.

Examples

(i) Any metric space (X,d) is a topological space where T_d, the topology for X induced by the metric d, is defined by agreeing that G shall be declared as open whenever each x in G is contained in an open ball entirely in G, i.e.

Æ Ì G Í X is open in (X, Td) Û "x Î G, $rx > 0 such that x Î Brx(x) Í G.


(ii) The following is a special case of (i), above. Let R be the set of real numbers and let Á be the usual (metric) topology defined by agreeing that

Æ Ì G Í X is open in (R, Á) (alternatively, Á-open) Û "x Î G, $rx > 0 such that (x-rx, x+rx) Ì G.


(iii) Define T₀ = {Æ, X}
(iv) Define D = { G Í X: G Í X }
(v) For any non-empty set X, the family C = { G Í X: G = Æ or X\G is finite} is a topology for X called the cofinite topology.
(vi) For any non-empty set X, the family L = { G Í X: G = Æ or X\G is countable} is a topology for X called the cocountable topology.

1.1.2  Neighbourhoods

Occasionally, arguments can be simplified when the sets involved are not ``over-described''. In particular, it is sometimes suffices to use sets which contain open sets but are not necessarily open. We call such sets neighborhoods.

Definition 2 Given a topological space (X, T) with x Î X, then N Í X is said to be a (T)-neighbourhood of x Û $ open set G with x Î G Í N.

It follows then that a set U Í X is open iff for every x Î U, there exists a neighbourhood N_x of x contained in U. (Check this!)

Lemma 1 Let (X, T) be a topological space and, for each x Î X, let N(x) be the family of neighbourhoods of x. Then

(i) U Î N(x) Þ x Î U.
(ii) N(x) is closed under finite intersections.
(iii) U Î N(x) and U Í V Þ V Î N(x).
(iv) U Î N(x) Þ $W Î N(x) such that W Í U and W Î N(y) for each y Î W.

Proof Exercise!

Examples

(i) Let x Î X, and define T_x = { Æ, {x}, X }. Then T_x is a topology for X and V Í X is a neighbourhood of x iff x Î V. However, the only nhd of y Î X where y ¹ x is X itself
(ii) Let x Î X and define a topology Á(x) for X as follows:

Á(x) = { G Í X: x Î G} È{ Æ}.

Note here that every nhd of a point in X is open.
(iii) Let x Î X and define a topology E(x) for X as follows:

E(x) = { G Í X: x \not Î G } È{X}.

Note here that {y} is open for every y ¹ x in X, that {x,y} is not open, is not a nhd of x yet is a nhd of y.

In fact, the only nhd of x is X.

1.1.3  Bases and Subbases

It often happens that the open sets of a space can be very complicated and yet they can all be described using a selection of fairly simple special ones. When this happens, the set of simple open sets is called a base or subbase (depending on how the description is to be done). In addition, it is fortunate that many topological concepts can be characterized in terms of these simpler base or subbase elements.

Definition 3 Let (X,T) be a topological space. A family B Í T is called a base for (X,T) if and only if every non-empty open subset of X can be represented as a union of a subfamily of B.

It is easily verified that B Í T is a base for (X, T) if and only if whenever x Î G Î T, $B Î B such that x Î B Í G.

Clearly, a topological space can have many bases.

Lemma 2 If B is a family of subsets of a set X such that

(B1) for any B₁, B₂ Î B and every point x Î B₁ ÇB₂, there exists B₃ Î B with x Î B₃ Í B₁ ÇB₂, and
(B2) for every x Î X, there exists B Î B such that x Î B,

Conversely, any base B for a topological space (X,T) satisfies (B1) and (B2).

Proof (Exercise!)

Definition 4 Let (X,T) be a topological space. A family S Í T is called a subbase for (X,T) if and only if the family of all finite intersections Ç_{i = 1}^kU_i, where U_i Î S for i = 1,2,¼,k is a base for (X,T).

Examples

(i) In any metric space (X,d), { B_r(x):x Î X, r > 0 } forms a base for the induced metric topology T_d on X.
(ii) For the real line R with its usual (Euclidean) topology, the family {(a,b): a,b Î Q, a < b } is a base.
(iii) For an arbitrary set X, the family { {x}:x Î X } is a base for (X, D).
(iv) The family of all `semi-infinite' open intervals (a, ¥) and (-¥,b) in R is a subbase for (R, Á).

1.1.4  Generating Topologies

From the above examples, it follows that for a set X one can select in many different ways a family T such that (X,T) is a topological space. If T₁ and T₂ are two topologies for X and T₂ Í T₁, then we say that the topology T₁ is finer than the topology T₂, or that T₂ is coarser than the topology T₁. The discrete topology for X is the finest one; the trivial topology is the coarsest.

If X is an arbitrary infinite set with distinct points x and y, then one can readily verify that the topologies Á(x) and Á(y) are incomparable i.e. neither is finer than the other.

By generating a topology for X, we mean selecting a family T of subsets of X which satisfies conditions (T1)-(T3). Often it is more convenient not to describe the family T of open sets directly. The concept of a base offers an alternative method of generating topologies.

Examples

• [Sorgenfrey line] Given the real numbers R, let B be the family of all intervals [x,r) where x,r Î R, x < r and r is rational. One can readily check that B has properties (B1)-(B2). The space R_s, generated by B, is called the Sorgenfrey line and has many interesting properties. Note that the Sorgenfrey topology is finer than the Euclidean topology on R. (Check!)
• [Niemytzki plane] Let L denote the closed upper half-plane. We define a topology for L by declaring the basic open sets to be the following:

  
  (I) the (Euclidean) open discs in the upper half-plane;
  (II) the (Euclidean) open discs tangent to the `edge' of the L, together with the point of tangency.

  Note If y_n ® y in L, then

  
  (i) y not on `edge': same as Euclidean convergence.
  (ii) y on the `edge': same as Euclidean, but y_n must approach y from `inside'. Thus, for example, y_n = ([1/n],0) \not ® (0,0)!

1.1.5  New Spaces from Old

A subset of a topological space inherits a topology of its own, in an obvious way:

Definition 5 Given a topological space (X, T) with A Í X, then the family T_A = {A ÇG:G Î T } is a topology for A, called the subspace (or relative or induced) topology for A. (A, T_A) is called a subspace of (X, T).

Example
The interval I = [0,1] with its natural (Euclidean) topology is a (closed) subspace of (R, Á).

Warning: Although this definition, and several of the results which flow from it, may suggest that subspaces in general topology are going to be `easy' in the sense that a lot of the structure just gets traced onto the subset, there is unfortunately a rich source of mistakes here also: because we are handling two topologies at once. When we inspect a subset B of A, and refer to it as 'open' (or 'closed', or a 'neighbourhood' of some point p .... ) we must be exceedingly careful as to which topology is intended. For instance, in the previous example, [0,1] itself is open in the subspace topology on I but, of course, not in the 'background' topology of R. In such circumstances, it is advisable to specify the topology being used each time by saying T-open, T_A-open, and so on.

1.2  Closed sets and Closure

• define closed sets in a general topological space and
• examine the related notion of the closure of a given set.

1.2.1  Closed Sets

Definition 6 Given a topological space (X,T) with F Í X, then F is said to be T-closed iff its complement X \F is T-open.

From De Morgan's Laws and properties (T1)-(T3) of open sets, we infer that the family F of closed sets of a space has the following properties:

(F1) X Î F and Æ Î F
(F2) F is closed under finite union
(F3) F is closed under arbitrary intersection.

Sets which are simultaneously closed and open in a topological space are sometimes referred to as clopen sets. For example, members of the base B = { [x,r):x, r Î R, x < r, r rational } for the Sorgenfrey line are clopen with respect to the topology generated by B. Indeed, for the discrete space (X, D), every subset is clopen.

1.2.2  Closure of Sets

Definition 7 If (X,T) is a topological space and A Í X, then

Evidently, [`A]<sup>T</sup> (or [`A] when there is no danger of ambiguity) is the smallest closed subset of X which contains A. Note that A is closed Û A = [`A].

Lemma 3 If (X, T) is a topological space with A, B Í X, then

(i) [`(Æ)] = Æ
(ii) A Í [`A]
(iii) [`([`A])] = [`A]
(iv) [`(AÈB)] = [`A] È[`B].

Proof Exercise!

Theorem 1 Given a topological space with A Í X, then x Î [`A] iff for each nhd U of x, U ÇA ¹ Æ.

Proof

$\Rightarrow$: Let x Î [`A] and let U be a nhd of x; then there exists open G with x Î G Í U. If U ÇA = Æ, then G ÇA = Æ and so A Í X \G Þ [`A] Í X \G whence x Î X \G, thereby contradicting the assumption that U ÇA = Æ.
$\Leftarrow$: If x Î X \[`A], then X \[`A] is an open nhd of x so that, by hypothesis, (X \[`A]) ÇA ¹ Æ, which is a contradiction (i.e., a false statement).

Examples

(i) For an arbitrary infinite set X with the cofinite topology C, the closed sets are just the finite ones together with X. So for any A Í X,

_ A   = ì í î A if A is finite X if A is infinite

Note that any two non-empty open subsets of X have non-empty intersection.


(ii) For an arbitrary uncountable set X with the cocountable topology L, the closed sets are the countable ones and X itself. Note that if we let X = R, then [`[0,1]] = R! (In the usual Euclidean topology, [`[0,1]] = [0,1].)
(iii) For the space (X, T_x) defined earlier, if Æ Ì A Í X, then

_ A   = ì í î X if x Î A X \{x} if x \not Î A


(iv) For (X, Á(x)) with A Í X,

_ A   = ì í î A if x \not Î A X if x Î A


(v) For (X, E(x)) with Æ Ì A Í X,

_ A   = ì í î A if x Î A AÈ{x} if x \not Î A


(vi) In (X, D), every subset equals its own closure.

1.3  Continuity and Homeomorphism

Because many properties of spaces are preserved by continuous functions, spaces related by a bijection (one-to-one and onto function) which is continuous in both directions will have many properties in common. These properties are identified as topological properties. Spaces so related are called homeomorphic.

1.3.1  Continuity

Definition 8 Let (X, T) and (Y, S) be topological spaces; a mapping f:X® Y is called continuous iff f^-1(U) Î T for each U Î S i.e. the inverse image of any open subset of Y is open in X.

Examples

(i) If (X, D) is discrete and (Y, S) is an arbitrary topological space, then any function f:X ® Y is continuous!

Again, if (X, T) is an arbitrary topological space and (Y, T₀) is trivial, any mapping g:X ® Y is continuous.
(ii) If (X, T), (Y, S) are arbitrary topological spaces and f:X ® Y is a constant map, then f is continuous.
(iii) Let X be an arbitrary set having more than two elements, with x Î X. Let T = I(x), S = T_x in the definition of continuity; then the identity map id_X:X ® X is continuous. However, if we interchange T with S so that T = T_x and S = Á(x), then id_X:X ® X is not continuous! Note that id_X:(X, T₁) ® (X, T₂) is continuous if and only if T₁ is finer than T₂.

Theorem 2 If (X₁, T₁), (X₂, T₂) and (X₃, T₃) are topological spaces and h:X₁ ® X₂ and g:X₂ ® X₃ are continuous, then g°h:X₁ ® X₃ is continuous.

Proof Immediate.

There are several different ways to 'recognise' continuity for a mapping between topological spaces, of which the next theorem indicates two of the most useful apart from the definition itself:

Theorem 3 Let f be a mapping from a topological space (X₁, T₁) to a topological space (X₂, T₂). The following statements are equivalent:

(i) f is continuous,
(ii) the preimage under f of each closed subset of X₂ is closed in X₁,
(iii) for every subset A of X₁, f([`A]) Í [`f(A)].

Proof It is easy to see that (i) implies (ii). Assuming that (ii) holds, apply it to the closed set [`f(A)] and (iii) readily follows. Now if (iii) is assumed and G is a given open subset of X₂, use (iii) on the set A = X₁ \f^-1(G) and verify that it follows that f^-1(G) must be open.

1.3.2  Homeomorphism

Definition 9 Let (X, T), (Y, S) be topological spaces and let h:X ® Y be bijective. Then h is a homeomorphism iff h is continuous and h^-1 is continuous. If such a map exists, (X, T) and (Y, S) are called homeomorphic.

Such a map has the property that

Example
For every space (X, T), the identity mapping id_X:X ® X is a homeomorphism.

A property of topological spaces which when possessed by a space is also possessed by every space homeomorphic to it is called a topological invariant. We shall meet some examples of such properties later.

One can readily verify that if f is a homeomorphism, then the inverse mapping f^-1 is also a homeomorphism and that the composition g °f of two homeomorphisms f and g is again a homeomorphism. Thus, the relation `X and Y are homeomorphic' is an equivalence relation.

In general, it may be quite difficult to demonstrate that two spaces are homeomorphic (unless a homeomorphism is obvious or can easily be discovered). For example, to verify that (R, Á) is homeomorphic to (0,1) with its induced metric topology, it is necessary to demonstrate, for instance, that h:(0,1) ® R where h(x) = [(2x-1)/(x(x-1))] is a homeomorphism.

It is often easier to show that two spaces are not homeomorphic: simply exhibit an invariant which is possessed by one space and not the other.

Example
The spaces (X, Á(x)) and (X, E(x)) are not homeomorphic since, for example, (X, Á(x)) has the topological invariant `each nhd is open' while (X, E(x)) does not.

1.4  Additional Observations

Definition 10 A sequence (x_n) in a topological space (X, T) is said to converge to a point x Î X iff (x_n) eventually belongs to every nhd of x i.e. iff for every nhd U of x, there exists n₀ Î N such that x_n Î U for all n ³ n₀.

Caution

We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence). However, for general topological spaces, sequential convergence fails. We illustrate:

(i) Limits are not always unique. For example, in (X, T₀), each sequence (x_n) converges to every x Î X.
(ii) In R with the cocountable topology L, [0,1] is not closed and so G = (-¥,0) È(1,¥) is not open - yet if x_n ® x where x Î G, then Assignment 1 shows that x_n Î G for all sufficiently large n. Further, 2 Î [`[0,1]]^L, yet no sequence in [0,1] can approach 2. So another characterisation fails to carry over from metric space theory.

Finally, every L-convergent sequence of points in [0,1] must have its limit in [0,1] - but [0,1] is not closed (in L)!

Hence, to discuss topological convergence thoroughly, we need to develop a new basic set-theoretic tool which generalises the notion of sequence. It is called a net - we shall return to this later.

Definition 11 A topological space (X, T) is called metrizable iff there exists a metric d on X such that the topology T_d induced by d coincides with the original topology T on X.

The investigations above show that (X, T₀) and (R, L) are examples of non-metrizable spaces. However, the discrete space (X,D) is metrizable, being induced by the discrete metric