1701354 Introduction to Topology – Nguyen
1-24-06
Class Handout
CONTINUITY, OPEN INTERVALS AND CONVERGENCE
I. Continuity and Open Intervals
Definition A. We say that a function f : \ → \ is continuous at a if for every ε > 0 there exists
a value δ > 0 such that f ( x) − f (a) < ε for all x satisfying x − a < δ .
Example: Definition A given above defines continuity using the notion of distance since it
employs absolute value inequalities to measure closeness. However, this is not necessary,
especially if we want to extend continuity to functions defined on arbitrary sets where these
notions do not exist. See if you can come up with more set-theoretic definitions, called B
through E below, that are equivalent to Definition A. In each case, write out your new definition
and discuss your reasons for adopting it, including examples and counterexamples where
appropriate):
Definition B: Try to modify Definition A so that absolute value inequalities are replaced by open
intervals centered at the point of continuity. Write your new definition below:
Definition C: Try to modify Definition B so that the notion of point elements (‘ ∈ ’ notation) is
replaced by the notion of subsets (‘ ⊆ ’ notation). Write your new definition below:
Definition D: Try to modify Definition C so that it allows for open intervals not necessarily
centered at the point of continuity. Write your new definition below:
Definition E: Try to modify Definition D so that it describes continuity in terms of open
intervals and inverse images (see note below). Write your new definition below:
Note: Let f : \ → \ be a function and V ⊆ \ a subset. Then the inverse image of V is defined
to be the subset f −1 (V ) , e.g. if f ( x ) = 3 x , then the inverse image of [0,1] is f −1 ([0,1]) = [0,1/ 3] .
II. Continuity and Convergence
To intuitively describe continuity, we normally use phrases like ‘f should get arbitrarily close to
f ( a ) as x gets arbitrary close to a’. This suggests treating ‘x’ as a sequence {xn } converging to
a, and { f ( xn )} as a sequence converging to f ( a ) .
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1701354 Introduction to Topology – Nguyen
1-24-06
Example: Let’s construct a definition for continuity in terms of sequences.
Definition F: Try to modify Definition A so that continuity is defined in terms of sequences.
Write your new definition below:
Example: Prove that f ( x) = x 2 is continuous at x = 2 based on Definition F.
Exercise 1: Prove rigorously that Definitions A and F are equivalent, i.e. each definition implies
the other.
III. Continuity on Subsets of \
Let D be a subset of \ and f : D → \ be a function on D. How can we define f to be
continuous on D instead of all of \ ? An example is f : ` → \ defined by f ( x ) = 1/ x and
whose domain is restricted to the positive integers.
Example: Let’s construct a definition for continuity restricted to subsets:
Definition A’: Modify Definition A to describe continuity at a point for a function defined on a
subset D ⊆ \ . Write your new definition below:
Example: Discuss the continuity of f : _ → \ defined by
⎧⎪1 if x ∈ _ and x < 2
f ( x) = ⎨
⎪⎩2 if x ∈ _ and x > 2
Exercise 2: Discuss the continuity of f : ` → \ defined by f ( x ) = x . Be sure to use
Definition A’ to rigorous argue where f is continuous. Hint: Choose any δ < 1 .
IV. Topologically (Continuously) Equivalent Subsets of \ :
How should we mathematically answer the question, “When are two sets equivalent to each
other?” The answer depends on the property that you want to compare or preserve. For
example, let’s consider the notion of size:
1. Cardinality (size):
Definition: We say that A and B have the same cardinality if there exists a bijection φ : A → B ,
i.e. a mapping f that is one-to-one and onto. This establishes a one-to-one correspondence
a ↔ b = φ ( a ) between the elements of A and those of B.
Example: Prove that the open unit interval (0, 1) and the open interval (0, 3) have the same
cardinality. Prove or disprove: (0, 1) has the same cardinality as the real line \ .
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1701354 Introduction to Topology – Nguyen
1-24-06
Let’s next consider boundary. For example, the open unit interval (0, 1) and the closed unit
interval [0,1] have the same cardinality, but not the same boundary. This is distinguished by the
fact that any bijection between them can never be continuous (a proof will be given later in this
course). This leads to us define the notion of topological equivalence between sets by requiring
both the bijection and its inverse to be continuous.
2. Topology (‘boundary’ structure)
Definition: We say that A and B are homeomorphic (or have the same topology or topologically
equivalent) if there exists a homeomorphism φ : A → B , i.e. a bijection φ such that both φ and
φ −1 are continuous maps (based on Definition E say).
Example: Prove that the open unit interval (0, 1) is homeomorphic to the open interval (0, 3).
Exercise 3: Prove that the closed intervals [0, 3] and [0, 9] have the same topology by finding
two different homeomorphisms φ and showing that each is indeed a homeomorphism.
Exercise 4: Consider a map φ that sends the interval (0, 2π ] ⊂ \ to the unit circle S 1 ⊂ ^
(complex plane):
φ : (0, 2π ] → S 1
t 6 eit = cos t + i sin t
Here, S 1 = {z = x + iy ∈ ^ : z = x 2 + y 2 = 1} . Show that φ is a bijection, but not a
homeomorphism. Note: By replacing ^ with \ 2 , we can view φ as a mapping from
(0, 2π ] ⊂ \ to S 1 ⊂ \ 2 and defined by φ (t ) = (cos t , sin t ) .
V. Supplementary Questions
Question 1: Is it really necessary to restrict the definition of continuity to open intervals?
Exercise 5: Discuss whether or not continuity can be described in terms of more general sets. If
yes, then describe the more general kinds of open sets that can be used instead of intervals by
defining an open set. Then revise Definition E appropriately. Hint: An open set is one that
contains arbitrarily small open intervals.
Question 2: Is it really necessary to consider open intervals instead of closed intervals in defining
continuity?
Example: Discuss whether or not continuity can be defined using closed intervals instead of open
intervals. This means relaxing the absolute value inequalities so that they do not have to be strict
and replacing f ( x) − f (a) < ε and x − a < δ by f ( x) − f (a) ≤ ε and x − a ≤ δ , respectively.
If yes, then give a counterexample. If not, then give an equivalent definition in terms of closed
intervals. Illustrate with examples and counterexamples where appropriate.
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