Continuity and Convergence
4.1. What You Need to Know 1. Continuity
DEFINITION. Let (X, T) and (Y, T′) be two topological spaces. We say that the function f: (X, T)
→ (Y, T′) is continuous if
The next results gives a characterization of continuity:
THEOREM. (cf. Exercise 4.3.3). Let (X, T) and (Y,T′) be two topological spaces. Let f : (X, T) → (Y, T′) be a function. Then the following are equivalent:
(1) f is continuous;
(2) for any
(3) for any closed element V in T′, f–1(V) is closed in T.
DEFINITION. Let (X, T) and (Y,T′) be two topological spaces. We say that the bijective mapping f : (X, T) → (Y, T′) is a homeomorphism if f and f–1 are both continuous.
We say that two topological spaces are homeomorphic if there exists a homeomorphism between them.
EXAMPLES.
(1) Let f : R → (–1, 1) be defined by:
Then f is a homeomorphism (see Exercise 4.3.7).
(2) Every isometry is a homeomorphism.
(3) Any two open intervals in standard R are homeomorphic (see Exercise 4.3.7).
REMARK. Let X and Y be two topological spaces. Let f : X → Y be a function. Then it is clear that f is a homeomorphism iff the following three conditions hold:
(1) f is a bijection;
(2) for any open U in Y, f–1(U) is open in X;
(3) for any open V in X, f (V) is open in Y.
DEFINITION. Let X and Y be two topological spaces. Let f : X → Y be a function. Then (1) f is said to be open if for any open set U in X, f (U) is open in Y.
(2) f is said to be closed if for any closed set V in X, f (V) is closed in Y.
PROPOSITION. An open, continuous and bijective map is a homeomorphism.
DEFINITION. A topological property is a property shared by a given topological space and any other topological space homeomorphic to it.
EXAMPLES.
(1) We saw above that R is homeomorphic to (–1, 1). Thus two consequences arise:
(a) Boundedness is not a topological property (see also Exercise 4.3.5);
(b) The length is not a topological property either
( 2) Closedness is not a topological property either. In the usual topology, the closed R is homeomorphic to the open via the function "ln".
REMARK. Throughout this book, many examples of topological properties will be met.
DEFINITION. Let f : X → Y be a function where X and Y are two topological spaces. The graph of f, denoted by Gf, is defined as:
4.1.2. Convergence.
DEFINITION. Let (X, T) be a topological space. A sequence (xn) in X converges to x ∈ X if:
EXAMPLE. Let X = {1,2,3} be equipped with the topology:
T = {∅, {2}, {1, 2}, {2, 3}, X}.
Then the sequence defined by xn = 2 converges to 2. It also converges to 1 and 3. Thus a sequence in an arbitrary topological space can converge to more than one limit.
REMARK. One has to be careful with sequences in topological spaces. The topology which equips a set may give us "surprises". For example, sequences like xn = n or (–1)n may converge.
In a metric space we have the following definition (equivalent to the previous one in this setting) DEFINITION. Let (X, d) be a metric space. A sequence (xn) in X converges to x ∈ X if:
EXAMPLE. In a discrete metric space, the only convergent sequences are the eventually constant ones.
THEOREM. In a separated (Hausdorff) topological space, a sequence cannot converge to two different points.
Since any metric space is Hausdorff (Exercise 3.5.10), the previous result implies COROLLARY. In a metric space, if a sequence converges, then its limit is unique.
We now give a fundamental and very practical result in metric spaces:
REMARK. Doubtlessly, the second result in the previous theorem can also be used to show that a given set is (or is not) open. See Exercises 4.3.21 & 4.3.22.
4.1.3 Sequential Continuity. We now come to a practical definition of continuity. But, in a general setting, it not as strong as the definition of continuity seen in the beginning of this chapter.
DEFINITION. Let X and Y be two topological space and let f : X → Y be a function. Then we say that f is sequentially continuous at x ∈ X if for any convergent sequence (xn) to x, (f(xn)) converges to f(x).
As usual, if f is sequentially continuous at each x ∈ X, then we say that f is sequentially continuous on X.
REMARK. Continuity implies sequential continuity but not vice versa. See Exercise 4.3.16.
However, the two notions match in a metric space and we have
PROPOSITION (For a proof see Exercise 4.3.16). Let (X, d) and (Y, d′) be two metric spaces and let f : (X, d) → (Y, d′) be a function. Then f is continuous on X iff it is sequentially continuous on X.
4.2. True or False: Questions
QUESTIONS. Comment on the following questions/statements and indicate those which are false and those which are true when this applies. Justify your answers.
(1) The identity function between two topological spaces is always continuous.
(2) Let X and Y be two topological spaces. If f is a continuous function from X into Y, then for (5) The sequence converges to zero.
(6) Criticize the following proof: In the co-finite topology on R, the sequence converges to any point in R. To prove it, assume that for all x ∈ R. This means that
Hence ∈ Uc for all but finitely many n and this is a contradiction since Uc is finite!
(7) Find fault with the following reasoning: In usual R, consider the set
Then A is not closed as (xn) does not converge and hence it cannot have a limit belonging to A.
(8) In the usual topology, let f : → R be such that f(x) = ln x. Then f is continuous, but [0, 1] is closed in R and yet its preimage is not closed in R. Is there anything wrong with that?
(9) If f is continuous and –f is well-defined, then –f is continuous.
(10) Let X be a topological space and let (xn) be a convergent sequence in X. Then X is Hausdorff (xn) has a unique limit.
(11) Let X be a topological space. Endow R with its usual topology. Let f : X → R be some function. If we come to show that f–1((a, b)) (a and b real numbers with a < b) is open in X, then f is continuous.
(12) When are two sets homeomorphic? When are they not homeomorphic?
(13) Let X and Y be two topological spaces and let f be a continuous mapping from X into Y.
Assume that A ⊂ X is dense. Then f(A) is dense in f(X).
(14) Separability is a topological property.
(15) A bijective mapping between two topological spaces is continuous.
(16) An open, closed and bijective mapping between two topological spaces is continuous.
(17) Let X be a topological space and let A ⊂ X. Let f : X → X be a bijective mapping. Then f is a homeomorphism iff
(18) Every continuous bijection is a homeomorphism.
(19) Every continuous and open bijection is a homeomorphism.
(20) Hausdorffness is a topological property.
Exercise 4.3.1. Let X and Y be two topological spaces. Let f : X → Y be a function. If f continuous in the following cases?
(1) f(x) = x, X is the indiscrete topology and Y = X equipped with the discrete topology;
(2) f(x) = ex, X = (R, | · |) and Y = R endowed with the discrete topology.
(3) f(x) = x2, X = (R,T) and Y = (R, | · |) where
(4) f arbitrary X discrete and Y arbitrary;
Exercise 4.3.3. Let X and Y be two topological spaces. Let f : X → Y be a function. Show that f is continuous
Exercise 4.3.4. Let X be an uncountable set. We endow X with the co-finite topology (see Exercise 3.3.22) and denote it by T. We also equip X with the "co-countable topology" (see Exercise 3.3.24) and we denote it by T′. Let f : T′ → T be defined for any x ∈ X by f (x) = x. Is f a
(2) Deduce that boundedness is not a topological property.