Basic Topology on Metric Spaces

Countable Sets

Theorem 1. If there is a surjection , we have . If there is an injection , we have .

Definition 2 [Equivalence between sets]
If there is a bijection form set to set , we say they have the same cardinality number. In short, they are equivalent. We write .

Definition 3. [Finite, Countable]

Let represents set {1, 2, 3, ..., n}. If set satisfies for some , is finite. If is not finite, is infinite.
Let represents the set of all positive integers. If set satisfies , is countable. A set is uncountable if it is neither finite nor countable.¹
is at most countable if it is countable or finite.

Remark 4. A finite set can't be equivalent to one of its proper subsets, but some infinite sets can.

Theorem 5. Every infinite subset of a countable set is countable.

Theorem 6. An at most countable union of at most countable sets are countable.

Corollary 7. The set of rational numbers are countable, but the set of real number are not.

Corollary 8. The set of infinite sequences is uncountable.

Corollary 9. for all .

Metric Spaces

Definition 10. [Metric space, distance function]

A combination of a set and a map , which is written as , is called a metric space if for :
1. , if ; ;
2. ;
3. for .
Every function such that satisfies the three properties are called distance function.

Definition 11. [Segment, Interval]

The set of rational numbers such that for any given is called the segment .
The set of rational numbers such that for any given is called the interval .

Definition 12. [Neighborhood, Limit point, Interior point]

For any given metric space and set , we have following definitions:

A neighborhood of a point with radius is a set .
A point is a limit point of set if , we have . The set of all limit points of set is written as .
A point is a interior point of set if .

Definition 13. [Closed, Open, Closure]

For any given metric space and set , we have following definitions:

Set is open if all limit points of is in . That is, .
Set is closed if every point of is an interior point of .
The closure of a set is the union of its limit points and itself. That is, .

Definition 14. [Complement, Perfect, Bounded, Dense]

For any given metric space and set , we have following definitions:

The complement of is .
is perfect if .
is bounded if there is a real number and a point s.t. .
is dense in if .

Corollary 15. Every neighborhood is an open set.

Corollary 16. , set is infinite. Therefore, a finite set has no limit point.

Theorem 17. . That is, the complement of a union of sets is the intersection of the complements of each set.

Theorem 18. A set is open if and only if its complement is closed. is closed if and only if its complement is open.

Theorem 19. The relationship of the union/intersection of sets between its openness and closedness:

For a finite collection of open sets, both its intersection and union are open;
For a finite collection of closed sets, both its intersection and union are closed;
For a infinite collection of open sets, only its union is guaranteed to be open;
For a infinite collection of closed sets, only its intersection is guaranteed to be closed.

Theorem 20. If is a metric space and , then:

is closed;
if and only if is closed;
such that and is closed, .

Corollary 21. Let be a nonempty set of real numbers which is bounded above, then sup , and sup if E is closed.

Definition 22. [Open relative]

For any subset and of metric space such that , we say is open relative to if for each point , there is a radius such that and .
It's clear to see that the definition is equivalent to there is an open subset such that = .

Compact

Definition 23. [Open cover, Subcover, Compact, Relatively Compact]

An open cover of is a collection of open subsets of , such that . That is, is "covered" by .
A subcover of an open cover is the subset of , while still "cover" the set , that is, while .
is compact in if for every open cover , there is a subcover.
Suppose , then is compact in if and only if is compact in , which can also be written as is compact relative to .

Theorem 24. Compact subset and metric space are closed and bounded.

Theorem 25. Every closed subset of compact set is compact.

Corollary 26. If is closed and is compact, then is compact too.

Theorem 27. For any given collection of compact sets such that for every finite subset , its intersection is not empty, then is not empty too.

Corollary 28. For a chain of nonempty compact sets in such that is not empty for any , then is not empty too.

Theorem 29. For any compact set , every infinite subset of has a limit point in .

There are different definitions for countable. Some people say if a set is finite, it is also countable, and infinite countable set it "infinitely countable". In the following paragraphs, we will mainly follow the definition from Rudin.↩︎