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Lecture 6 outline
FUNCTIONS
• Functions: A map ƒ: X → Y is a function. It assigns each element in X a unique
element in Y.
a) Key example: ƒ: X → R. Assigns each point in X a number.
b) Or, ƒ: X → Rn.
c) Example of a knot in and map ƒ: S1 → R3.
d) The Gauss map from a surface to S2.
HOMEOMORPHISMS
• Homeomorphism equivalence: Let X and Y be topological spaces.
a) Notion of equivalence: A map ƒ: X → Y is a homeomorphism when it is a
bijection and both ƒ and ƒ-1 map open sets to open sets.
b) The condition that X and Y be homeomorphic is an equivalence relation.
c) Basic question: What are the equivalence classes?
d) Are all spaces of the same cardinality equivalent?
1) Is R homeomorphic to R2?
2) Is R with the standard topology homeomorphic to R with the upper/lower limit
topologies? Note: Upper limit R is homeomorphic to lower limit R using the
map ƒ(t) = -t
3) Is R2 homeomorphic to S2? Is T2 homeomorphic to S2?
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4) Any two open intervals (either bounded or not) are homeomorphic.
5) Let K and K´ be knots in R3. Is R3−K homeomorphic to R3−K´. If not, what
distinguishes them?
6) When (if ever) is Πα∈J Xα with box topology homeomorphic to the same set
with the product topology?
Is every space homeomorphic to a subspace of Euclidean space? If not, which
topological spaces are?
DIGRESSION ON METRIC SPACES
• A distance function on a set X is a function d: X × X → X that obeys the following
rules:
a) d(x, y) = d(y, x) (it is symmetric)
b) d(x, y) ≥ 0 with equality if and only if x = y (positivity)
c) d(x, y) + d(y, z) ≥ d(x, z) (triangle inequality).
• Examples:
a) Euclidean space
b) Spaces of functions and sup norm.
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Metric topology: A basis for the topology consists of the ‘balls’ centered at the points
in X: A basis set is labeled by (x, ε) with x ∈ X and ε > 0: It is the set of x´ ∈ X such
that d(x´, x) < ε.
a) The triangle inequality guarantees that these balls satisfy the intersection property
that is needed to be a basis.
b) The topology is Hausdorff (because d(x, y) > 0 unless x = y.)
Standard topology on Rn is metric topology.
The subspace topology for a subset of a metric space is a metric topology
Basic question: Is every (Hausdorff) topology a metric topology? If not, what are the
necessary and sufficient conditions?
CONTINUITY
• The notion of a function being continuous (with respect to a given topology) brings
the open sets into the story as the open sets distinguish a certain subset of functions.
• This subset of functions are called continuous.
• Suppose X, Y are topological spaces: A function ƒ: X → Y is said to be continuous
when the following occurs: If O ⊂ Y is any open set, then ƒ-1(O) is an open set in X.
(Keep in mind that ƒ-1(O) is the set {x ∈ X: ƒ(x) ∈ O}.