Sequentially compact space

In mathematics, a topological space is sequentially compact if every sequence has a convergent subsequence. Metric spaces are sequentially compact if and only if they are totally bounded and complete, so compact. Therefore, subsets of exactly then sequentially compact ( and compact) when they are closed and bounded. There are topological space, which are compact and non- compact sequence, and rooms that are not sequentially compact, but is compact.

• 4.1 A compact Hausdorff space that is not sequentially compact
• 4.2 A sequence of compact space, which is not compact

Definitions

Convergent sequences in topological spaces

If a metric space, so converges to a sequence when

This means that the sequence iff to converge when there is a natural number for each positive (small ) number so that all the sequence elements have at a distance from which is less than.

In any topological spaces the environments take the place of the balls. If a topological space, then converges to a result if there is any environment from a so for all.

Sequential compactness

A topological space is called sequentially compact if every sequence contains a convergent subsequence.

Metric spaces

A metric space is sequentially compact if and only if it is compact. For a metric space is compact if it is totally bounded and complete.

Each episode is a metric space totally bounded, so contains a Cauchy sequence as a subsequence. Is he in addition complete, this sequence converges. A compact metric space is therefore sequentially compact. General erstabzählbare any compact space is sequentially compact.

If a metric space sequentially compact reversed so he must be totally bounded, since one could otherwise find and a sequence of points, each at a distance of, and would therefore have no convergent subsequence. The room must also be complete, since a convergent subsequence of a Cauchy sequence must have the same limit as the original sequence.

Properties follow compact spaces

A topological space is called countably compact if every sequence has a cluster point. Every sequentially compact space is countably compact. ( The converse is not true. ) In particular, each sequentially compact space is weakly countably compact and pseudo- compact, there is also every countably compact space.

Examples

A compact Hausdorff space that is not sequentially compact

The amount provided with the discrete topology, is compact, so by the theorem of Tychonoff is the set of all functions from the interval after, provided with the product topology compact, moreover, this space is Hausdorff.

That is provided with the product topology means that a sequence of functions converges if it converges pointwise.

This space is not sequentially compact:

A sequence of features that contains no convergent sequence can be defined as follows:

In the decimal notation in the dual analog system, the fractional part of a real number is an infinite sequence of zeros and ones.

The sequence is now defined as follows: is the -th decimal number.

A subsequence of a number can then be defined as follows. In the binary point representation has on the -th digit one, if just is and if is odd, one at the other places. This means that the sequence does not converge, because in point, the values ​​jump back and forth. The result can therefore have no convergent subsequence.

Since the space is compact, but has the consequence of a convergent subnet.

A sequence of compact space, which is not compact

The first uncountable ordinal number (ie the uncountable set of all countable ordinals ) is given by the relation ( the relation ) well-ordered and therefore carries the topology of this order.

Is now a sequence of ordinals, then the least ordinal with the property that only finitely many terms of the sequence greater than they are, is a limit point of this sequence and the sequence can be thinned to a convergent sequence. The space is therefore countably compact and sequentially compact.

The family of open sets covers the set of all countable ordinals. A finite subfamily contains only countably many elements of. is therefore not compact.

The fact that the amount is not compact, is because it does not contain the limit ordinal. But this is not the limit of a countable sequence, but only the limit of an uncountable network ( approximately given by all countable ordinal numbers in their natural order).