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 COMPACT, a tutorial by Scott W. Williams title and preface page 1: the beginnings page 2: a special example page3: functional separation page5: open covers References

1. the beginnings

The Cech Mathematician Bernard Bolzano did a number of remarkable things very early. In particular, in 1817, he extacted numbers from the notion of sequence [Kline1972], and gave early formulation of finite and infinite sets [Cantor 1883, Jarnik1981]. In the 1830's he showed that a function continuous on a closed interval is bounded, proved that a bounded sequence has a limit point [Bolzano1841, Jarnik1981], and gave the first example of a continuous nowhere differentiable function (usually attributed to Weierstrass 30 years later). The proof of these essentially led to the Bolzano-Weierstrass Theorem [Taylor 1982]:

1.1. THEOREM. Every infinite bounded subset of reals has a limit point.

A key to this theorem is an axiom implying that the real line has "no holes except at infinity": Call the set B. If B contains an infinite increasing sequence, then the least upper bound of the sequence is a limit point. The infinite decreasing sequence case is analogous. By a partitioning B, we see that a bounded set without monotone infinite sequences must be finite.

Before the 19th century, folks already knew that "small" polynomials attained their maxima and minima on closed intervals, but the Bolzano-Weierstrass Theorem led to what turns out to be one of the chief motivations for studying compactness, Weierstrass' theorem [Taylor 1982]:

1.2. THEOREM. Each function continuous on a closed subset of a closed interval attains its maximum.

The essence is that the continuous image of a closed and bounded set is closed and bounded. Today we know Theorem2 to be true for real-vlaued functions continuous on any compact space; however, theorems 1.1 and 1.2 suggest:

1.3. DEFINITION. COMPACT = "NO HOLES": This should be interpreted intuitively. The "proofs" we give in 1.4 are intuitive and need either to be fleshed out with the definition given in 1.5 or 5.1.

1.4. EXERCISES.

1. We recognize two possible kinds of holes - holes at infinity and holes nearby.

a. Loosely, compactness requires a kind of boundedness. is a hole of the non-negative integers {0,1,2,3,4 ...} is not compact. When the hole at is put in - considering as a point, the new object, , has no holes.

b. Compactness, loosely, requires a special kind of "closedness." 0 is a hole of the "convergent" sequence and the sequence, without its limit, is not compact. With its limit, it is compact. More generally, if (0,1] with its usual open sets is declared closed in some topology on the line, then [0,1] is not compact in that topology.

2. Removing the origin <0,0> from the unit disk also leaves a nearby hole:

Thus, is not compact.

3. Intutitively, "hole" implies some kind of an unending process without resolution. So intutively, finite sets are compact.

4. Suppose * is a hole of a closed subset F of a compact space X. As a there is stuff of F "close to" *, there is stuff of X "close to" *. But X is compact and * must be a point of X. Since F is a closed subset of X, it contains all points of X close to it. So, a closed subset of a compact set must be compact.

5. A far away hole: The space of all countable ordinal numbers has no holes approachable by a countable sequence, yet is not compact.

When a point has been put in the hole, the space is known as .

1.5. DEFINITION. A filter is a family of non-empty sets which satisfy the condition: If A,B, then there is a C contained in AB.

Here is a correct definition of "close to": converges to (clusters at) the point x provided each neighborhood of x contains (intersects) a member of .

Here is a correct definition of compact: A space is said to be compact provided each filter consisting of closed sets is contained in a converent filter; or equivalently, each filter consisting of closed sets clusters at some point.

This version of compactness was important since the 1940's, but is related to the first study of abstract compact spaces [Vietoris1921].

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 title and preface page 1: the beginnings page 2: a special example page5: open covers