7.12. GENERAL TOPOLOGICAL SPACES 163

Note how the target function f in the above example is not continuous even thougheach function in the sequence is. The nice thing about uniform convergence is that it takescontinuity of the functions in the sequence and imparts it to the target function. It does thisfor both continuity at a single point and uniform continuity. Thus uniform convergence isa very superior thing.

Theorem 7.11.11 Let fn : X → Y where (X ,d) ,(Y,ρ) are two metric spaces and supposeeach fn is continuous at x ∈ X and also that fn converges uniformly to f on X. Then f isalso continuous at x. In addition to this, if each fn is uniformly continuous on X , then thesame is true for f .

Proof: Let ε > 0 be given. Then

ρ ( f (x) , f (x̂))≤ ρ ( f (x) , fn (x))+ρ ( fn (x) , fn (x̂))+ρ ( fn (x̂) , f (x̂))

By uniform convergence, there exists N such that both ρ ( f (x) , fn (x)) and ρ ( fn (x̂) , f (x̂))are less than ε/3 provided n≥ N. Thus picking such an n,

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))

Now from the continuity of fn, there exists δ > 0 such that if d (x, x̂)< δ , then

ρ ( fn (x) , fn (x̂))< ε/3.

Hence, if d (x, x̂)< δ , then

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))<

2ε

3+

ε

3= ε

Hence, f is continuous at x.Next consider uniform continuity. It follows from the uniform convergence that if x, x̂

are any two points of X , then if n≥ N, then, picking such an n,

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))

By uniform continuity of fn there exists δ such that if d (x, x̂) < δ , then the term on theright in the above is less than ε/3. Hence if d (x, x̂)< δ , then ρ ( f (x) , f (x̂))< ε and so fis uniformly continuous as claimed.

7.12 General Topological SpacesIt turns out that metric spaces are not sufficiently general for some applications. This sec-tion is a brief introduction to general topology. In making this generalization, the propertiesof balls which are the conclusion of Theorem 7.1.4 on Page 135 are stated as axioms fora subset of the power set of a given set which will be known as a basis for the topology.More can be found in [83] and the references listed there.