Ten Ways of Looking at Real Numbers

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#2. Real Numbers as a Linear Order

Suslin Problem: Replace separability with the countable chain condition:

every collection of disjoint nontrivial closed intervals is at most countable.

Does this characterize the real numbers?

A counterexample is called a Suslin line

The existence of Suslin lines are independent of the axioms of ZFC set theory

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#2. Real Numbers as a Linear Order

The real numbers may be viewed as a space of branches of an infinite tree.

Trees are partial orders whose initial segments {x : X<p} are well-ordered. The branches are the maximal chains in the partial order.

Different infinite trees (Aronzsajn tree, Kurepa tree) give rise to different linear orders (Aronszajn line, Kurepa line).

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#3. Real Numbers as a Topological Space

A characterization of the usual topology of the real line:

If X is a regular, separable, connected, locally connected space, in which every point is a cut point, then X is homeomorphic to the real line.

A point p in a connected space X is a cut-point if X\{p} is disconnected.

Another characterization: Replace “regular” with “metrizable”.

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#3. Real Numbers as a Topological Space

The real numbers form a complete separable metric space, a “Polish space”. Also, it is perfect.

Other examples of perfect Polish spaces:

Cantor space: all sequences of 0’s and 1’s

Baire space: all sequences of natural numbers

Finite/countable products of perfect Polish spaces

Every perfect Polish space is Borel-isomorphic to the real numbers.

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#3. Real Numbers as a Topological Space

The real line is a one-dimensional topological manifold.

Classification of connected Hausdorff one-dimensional manifolds

the real line

the circle

the long line

the open long ray

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#4. Real Numbers as a Completion of the Rational Numbers

A valuation is a function from a field to the nonnegative real numbers with properties analogous to a norm or absolute value:

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#4. Real Numbers as a Completion of the Rational Numbers

Two valuations are equivalent if one is a power of the other.

Every valuation is equivalent to one which satisfies:

Such a valuation defines a metric on a field: the distance between a and b is

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