Course Review
Notions.
• Countable and uncountable sets.
• εδ-continuity for functions f : R → R.
• Metric spaces.
• Continuous maps between metric spaces: Sequential definition, εδdefinition, and the definition involving preimages of open sets.
• Open spheres and closed spheres.
• Open/closed sets in metric spaces.
• Interior, closure, boundary of a set.
• Limit points.
• Complete Metric Spaces. Cauchy sequences. Meager/first category
and residual/second category sets.
• Contraction mapping in complete metric spaces.
• Topological Spaces. Family of open sets. Neighborhoods.
• Comparison of Topologies.
• Discrete Topology.
• Trivial Topology.
• Subspace/induced topology.
• Sets: open, closed, boundary, interior, closure.
• Limit points, isolated points in topological spaces.
• Base of topology.
• Product topology: finite and infinite (countable) products. Box
topology.
• Projection mappings in the product spaces.
• Connected spaces. Path-connected spaces. Cut points.
• Compact Spaces. Open coverings, subcovers.
• Homeomorphisms. Homeomorphic topological spaces.
• Open and closed maps.
• Hausdorff spaces.
Main Examples.
• Q, R, Rd , S1 (unit circle), intervals.
• Finite topological spaces.
• Cartesian products with product topology.
• Middle third Cantor set ∼
= {0, 1}N (equipped with product topology).
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• Irrational numbers ∼
= N (equipped with product topology).
• Topologist’s sine curve (connected but not path-connected).
• Connected circles/intervals with different number of cut points.
• C([0, 1]) — the space of continuous functions on [0, 1] with metric
d(f, g) = supx∈[0,1] |f (x) − f (y)|.
• Torus ∼
= S1 × S1 .
• Cylinder ∼
= R × S1 .
• Co-finite topology. X = N and F consists of all co-finite sets.
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MAIN RESULTS.
• Distribution laws for unions/intersections of sets.
• The set of rational numbers is countable. The set of irrational numbers is uncountable.
Metric Spaces.
• Equivalent definitions of continuity for maps between metric spaces.
• Proof of the fact that C([0, 1]) is a metric space.
• (STATEMENT)
Rd is a metric space with respect to the metric
√∑
d(x̄, ȳ) = ( i (xi − yi )2 ). The Cauchy-Schwartz inequality.
• Unions of open sets are open; finite intersections of closed sets are
closed.
• the closure of a set is defined as the set itself with its limit points.
This is equivalent to being the smallest closed set containing the
original set. Ā = A ∪ ∂A; ∂A = Ā ∩ Ac . A is closed iff Ā = A.
• A is open iff Int(A) = A. Int(A) is the largest open subset of A.
• (STATEMENT) R is a complete metric space with respect to the
usual metric. Rn is a complete metric space.
• Y is a subset of a complete metric space X. Y is complete iff Y is
closed. For example, S1 is complete.
• Contraction mapping theorem/The fixed point theorem. Let f :
X → X be a contraction with contraction coefficient q. Then f has
a unique fixed point x∗ and d(f (n) (x0 ), x∗ ) ≤ |f (x0 ) − x0 |q n /(1 − q)
where x0 is an arbitrary point in X.
Topological Spaces.
• The criterion for a family B ⊂ F to be a base of F.
• The cylinder sets in finite product spaces form a basis for the product
topology. The projections in the product spaces are open.
• A map between topological spaces is a homeomorphism if and only
if it (as well as its inverse) sends elements of the bases onto open
sets.
• A = {x ∈ X|∀ nbhd U ∋ x, U ∩ A ̸= ∅}.
• A = A + {limit points of A}.
• The existence of the smallest topology containing any prescribed
family of sets. In other words, any family of sets can be used to
“generated” a topology.
• S1 × R is homeomorphic to a cylinder.
• Examples of maps that are (1) open; (2) closed; (3) open but not
closed; (4) closed but not open; (5) neither open but not closed.
• Example that projections need not be closed.
• The box topology on X = RN is strictly finer than the product
topology.
• The image of a connected (path-connected) space under a continuous
map is connected (path-connected).
• [a, b] is a connected space.
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• Compact subsets of a Hasdorff space are closed.
• (STATEMENT) The Heine-Borel theorem: [a, b] is compact.