There are many examples in general topology of defining a new topology on a set based on a given topology already defined on the set. For example, given a topology on , one can define a finer topology consisting of all sequentially open subsets of (based on the original topology ). Sequential spaces are precisely those spaces for which the original topology coincides with (see the post Sequential spaces, II). A related concept is the notion of k-spaces. We show that the compactly generated open sets form a finer topology and that k-spaces are precisely those spaces for which the compactly generated topology coincides with the original topology. We also give an external characterization of k-spaces, namely, those spaces that are quotient images of locally compact spaces. All spaces under consideration are Hausdorff.
Let be a space. We say is a compactly generated closed set in if is closed in for any compact . We say is a compactly generated open set in if is a compactly generated closed set in . The space is said to be a k-space if
is a compactly generated closed set in if and only if is a closed set.
The direction of the above statement always holds. So a space is a k-space if it satisfies the direction in the above statement. We also want to mention that in the above definition, “closed” can be replaced by “open”.
Suppose is the topology of the space . Then define as the set of all compactly generated open sets in . It can be easily verified that is a topology defined on the set and that is a finer topology than the original topology , i.e. . It follows that is a k-space if and only if .
Much of the discussion here mirrors the one in Sequential spaces, II. In a sequential space, the topology coincides with , the open sets generated by convergent sequences (a particular type of compact sets). In a k-space, the topology conincides with , the open sets generated by the compact sets. A sequential space is the quotient space of a topological sum of disjoint convergent sequences. Any k-space is the quotient space of a topological sum of disjoint compact sets.
We have the following theorem.
For any space , the following conditions are equivalent:
- is a k-space.
- is a quotient space of a locally compact space.
Proof. Let be a k-space. Let be the set of all compact subsets of . Let be the topological sum of all where each has the relative topology inherited from the space . Then is a locally compact space. There is a natural mapping we can define on onto . The space is a disjoint union of all compact subsets of . We can map each compact set onto the corresponding compact subset of by the identity map. Call this mapping where . We claim that the quotient topology generated by this mapping coincides with the original topology on .
Let be the given topology on and let be the quotient topology generated on the set . Clearly, . We need to show that . Let . Since is a k-space, if we can show that is a compactly generated open set, then .
Let be compact. We need to show that is open in . Since , is open in . Since is open in , is open in . It is also the case that is open in . We can consider as a subset of and as a subset of . As a subset of , we have . Thus is open in and .
Let be locally compact and let be a quotient map. We show that is a k-space. To this end, we show that if is a compactly generated closed set in , is closed in . Or equivalently, if is not closed in , then is not a compactly generated closed set in . Under the quotient map , is closed in if and only if is closed in .
Suppose is not closed in . Then is not closed in . Then there is . Let be open in such that and is compact. Then is compact. It follows that is not closed in . Note that and . However, . Thus is not a compactly generated closed set in .
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