Any topological space where there is a countable base at every point is said to satisfy the first axiom of countability or to be first countable. In this post we discuss several properties weaker than the first axiom of countability. All spaces under consideration are Hausdorff. Any first countable space satisfies each of the following conditions:
- If and then there is a sequence of points in such that the sequence converges to .
- The set is closed in if is sequentially closed in , which means that: if is a sequence of points in such that converges to , then .
- The set is closed in if this condition holds: if is compact, then is closed in .
- condition 1 are called Frechet spaces,
- condition 2 are called sequential spaces,
- condition 3 are called k-spaces.
All three of these conditions hold in first countable spaces. We have the following implications:
First countable Frechet Sequential k-space
This post is an introductory discussion of these notions. Each of the above implications is not reversible (see the section below on examples). After we discuss sequential spaces, we take a look at the behavior of these four classes of spaces in terms of whether the property can be passed onto subspaces (the property being hereditary) and in terms of images under quotient maps. For previous posts in this blog on first countable spaces and quotient spaces, see the links at the end of the post. Excellent texts on general topology are  and .
For a subsequent discussion on sequential space, see Sequential spaces, II.
The Forward Implications
First countable Frechet
Suppose and . Let be a local base at . Then choose and we have .
Let be sequentially closed in . Suppose is not closed in . Then there is such that . By Frechet, there is a sequence such that . Since is sequentially closed, , a contradiction. So any sequentially closed set in a Frechet space must be a closed set.
Suppose is not closed in . Since is sequential, there is a sequence such that and . The set is a compact set. Note that is not closed in . This shows that is a k-space.
Sequential spaces are ones in which the topology can be completely described by convergent sequences. Let be a space. Let . The set is said to be sequentially closed in if whenever we have a convergent sequence of points in , the sequential limit must be in . In other words, contains all the limits of the convergent sequences of points in . For , is sequentially open in if this condition holds: if is a sequence of points converging to some , then for all but finitely many . It can be verified that:
is sequentially open in if and only if is sequentially closed in .
A space is Frechet if and only if every subspace of is sequential.
Suppose is not a sequential space. Then there is such that is sequentially closed in but is not closed in . We show that is not Frechet. There is a point such that is a limit point of (in the subspace ) and . Since is sequentially closed, no sequence of points in can converge to (otherwise ).
Clearly, the point is also a limit point of with respect to the toplology of , i.e. with respect to . Since no sequence of points in can converge to , is not Frechet.
Suppose is not Frechet. Then there is such that and no sequence of points in can converge to . Consider the subspace . The set is sequentially closed in but is not closed in .
Every quotient space of a sequential space is always a sequential space.
Proof. Let be a sequential space. Let be a quotient map. We show that is a sequential space. Suppose that is sequentially closed. We need to show that is closed in . Because is a quotient map, is closed in if and only if is closed in . So we need to show is closed in .
Suppose for each and the sequence converges to . Then . Since the map is continuous, . Since is sequentially closed, . This means . Thus is sequentially closed in . Since is sequential, is closed in .
Every quotient space of a first countable space is sequential. Every quotient space of a Frechet space is sequential.
Example 1. First countable Frechet
The example is defined in the post An example of a quotient space, I. This is a non-first countable example. There is only one non-isolated point in the space. It is easy to verify it is a Frechet space.
Example 2. Frechet Sequential
We consider the space defined in the post An example of a quotient space, II. Note that the space is the quotient image of a first countable space. Thus is sequential by Corollary 3. Consider the subspace . Within , no sequence of points in can converge to the point . However, is a limit point of . Thus is not sequential. By Theorem 1, is not Frechet.
Example 3. Sequential k-space
Any compact space is a k-space. Let be the first uncountable ordinal. Then with the ordered topology is compact. Note that is sequentially closed but not closed. Thus is not sequential.
Example 4. A space that is not a k-space
This example is also defined in the post An example of a quotient space, II. Consider the subspace . Every compact subset of is finite. So is closed in for every compact . But is not closed.
Comments About Subspaces
Which of the four properties discussed here are preserved in subspaces? Or which of them are hereditary? It is fairly straightforward to verify that first countability is hereditary and so is the property of being Frechet. By Theorem 1, for any sequential space that is not Frechet has a subspace that is not sequential. Thus the property of being a sequential space is not hereditary. However, closed subspaces and open subspaces of a sequential space are sequential.
The property of being a k-space is also not hereditary. The space defined in An example of a quotient space, II is a sequential space (thus a k-space). Yet the subspace is not a k-space.
Comments About Quotient Mappings
Continuous image of a first countable space needs not be first countable. The other three properties (Frechet, sequential and k-space) are also not necessarily preserved by continuous mappings. A quick example is to consider any space that does not have any one of the four properties. Then consider with the discrete topology. Then the indentity map from onto is continuous.
Example 1 shows that the property of being first countable is not preserved by quotient map. Example 2 shows that the Frechet property is not preserved by quotient map. Theorem 2 shows that the property of being sequential space is preserved by quotient map. We have the following theorem about k-spaces under quotient map.
The spaces that are k-spaces are called compactly generated spaces. In a k-space, the closed sets and open sets are generated by compact sets. For example, for a k-space , is closed in if and only if is closed in for every compact . Let’s take another look at sequential spaces. The following definition is equivalent to the definition of sequential space given above:
is closed in if and only if is closed in for every compact of the form where the are a convergent sequence and is the sequential limit.
Thus the sequential spaces are compactly generated by a special type of compact sets, namely the convergent sequences.
Quotient images of k-spaces are always k-spaces.
Proof. Let be a k-space. Let be a quotient map. We wish to show that is a k-space. Suppose is not closed in . Since is a quotient mapping, is not closed in . Since is a k-space, there is a compact such that is not closed in . Let such that . We have just produced a compact set in such that is not closed in . Note that and is a limit point of . This implies that if is closed in for every compact , then must be closed in (i.e. is a k-space).
The discussion on sequential space continues with the post Sequential spaces, II.
Links to Previous Posts
An example of a quotient space, II
An example of a quotient space, I
The cardinality of compact first countable spaces, III
The cardinality of compact first countable spaces, II
The cardinality of compact first countable spaces, I
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.