# Hing Tong

**Hing Tong** was a mathematician whose name is attached to one of the standard results of general topology, the Katětov–Tong insertion theorem, which he proved in a 1952 paper in *Duke Mathematical Journal*<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup><sup> • </sup><sup>[2](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)</sup>. The theorem characterizes normal topological spaces by the existence of continuous functions squeezed between comparable pairs of semicontinuous functions, and it has been re-proved and generalized repeatedly since<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>. Beyond this result and a 1970 paper on topological expansions, the documented record of his life and work is thin: no obituary, departmental record, MathSciNet author profile, or genealogy entry is known<sup>[2](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)</sup>.

| Key fact | Detail |
|---|---|
| Signature result | Katětov–Tong insertion theorem: a space is normal iff every comparable pair of upper and lower semicontinuous functions admits a continuous function between them<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup> |
| Publication | "Some characterizations of normal and perfectly normal spaces", *Duke Math. J.* **19** (1952), 289–292, MR 50265<sup>[2](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)</sup> |
| Companion result | Miroslav Katětov proved the same theorem independently in *Fund. Math.* **38** (1951), 85–91, MR 50264<sup>[3](https://www.ams.org/journals/proc/1983-087-03/S0002-9939-1983-0684654-0/)</sup> |
| Special case | The theorem implies Urysohn's lemma, obtained by inserting between characteristic functions of a closed set and an open set containing it<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup> |
| Other publication | "Non-existence of certain topological expansions", *Annali di Matematica Pura ed Applicata* **86** (1970), answering a question of Edwin Hewitt<sup>[4](https://doi.org/10.1007/bf02415707)</sup> |
| Afterlife | At least a dozen distinct proofs and generalizations cataloged by 1993, with new versions appearing into the 2020s<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup><sup> • </sup><sup>[5](https://arxiv.org/abs/2001.08800)</sup> |

## The Katětov–Tong insertion theorem

The theorem connects two ways a space can fail to be well-behaved. A real-valued function on a topological space is *upper semicontinuous* if, for every real number a, the preimage of [a, ∞) is closed, and *lower semicontinuous* if the preimage of (−∞, a] is closed<sup>[6](https://www.dml.cz/bitstream/handle/10338.dmlcz/119148/CommentatMathUnivCarolRetro_41-2000-1_13.pdf)</sup>. Such functions can jump, unlike continuous ones, and the theorem says that a space is normal exactly when this gap can always be bridged.

**Statement.** A space X is normal if and only if whenever g is upper semicontinuous and h is lower semicontinuous with g ≤ h, there is a continuous f : X → ℝ such that g ≤ f ≤ h<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>.

The theorem matters because it converts the separation property of normality into a quantitative statement about functions: normality is precisely the condition under which semicontinuous upper and lower bounds can be met by a continuous function in between<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>.

**Two names, two proofs.** Katětov's proof appeared in "On real-valued functions in topological spaces", *Fundamenta Mathematicae* **38** (1951), pages 85–91 (MR 50264), and Tong's in "Some characterizations of normal and perfectly normal spaces", *Duke Mathematical Journal* **19** (1952), pages 289–292 (MR 50265)<sup>[3](https://www.ams.org/journals/proc/1983-087-03/S0002-9939-1983-0684654-0/)</sup><sup> • </sup><sup>[2](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)</sup>. The two proofs are independent and nearly simultaneous, which is why both mathematicians' names are attached<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>. A 2020 re-examination notes that neither Katětov's proof nor Tong's simplifies in the compact Hausdorff setting, so neither argument has an obvious technical short-cut the other lacks<sup>[5](https://arxiv.org/abs/2001.08800)</sup>.

## Place among the classical insertion theorems

The Katětov–Tong theorem sits in a family of results going back to Hans Hahn, who proved the insertion theorem for metrizable spaces, and [Jean Dieudonné](https://www.edgepedia.org/jean-dieudonne), who proved it for paracompact spaces<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>. Tong's version applies to all normal spaces<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>.

**Urysohn's lemma** is the special case in which the upper function is the characteristic function χ_K of a closed set K and the lower function is χ_U for an open set U containing K: inserting a continuous f with χ_K ≤ f ≤ χ_U is exactly the lemma's conclusion<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>.

**Dowker's theorem** strengthens the conclusion to strict inequality, g < f < h, and characterizes the spaces that are normal and countably paracompact; **Michael's theorem** characterizes perfect normality by requiring g(x) < f(x) < h(x) at every point where g(x) < h(x). Both follow directly from the Katětov–Tong insertion theorem<sup>[6](https://www.dml.cz/bitstream/handle/10338.dmlcz/119148/CommentatMathUnivCarolRetro_41-2000-1_13.pdf)</sup>. A 2014 survey-style article puts the relationship plainly: speaking ahistorically, the Katětov–Tong theorem is what remains of Michael's theorem once perfectness is dismantled<sup>[7](https://www.dml.cz/bitstream/handle/10338.dmlcz/144054/CzechMathJ_64-2014-3_9.pdf)</sup>.

## Later use and generalizations

The theorem has been re-proved and extended repeatedly. A 1993 paper by G. Kubiak catalogs at least a dozen distinct proofs and generalizations by Priestley, Jameson, Engelking, Michael, Blatter and Seever, Lane, Preiss and Vilímovský, Blair, Blair and Swardson, Kubiak, and Kotzé and Kubiak<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>. Kubiak's own strengthening inserts a pair of semicontinuous functions between a countable supremum of upper semicontinuous functions and a countable infimum of lower semicontinuous functions, yielding a characterization of completely normal spaces<sup>[1](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)</sup>.

Later work has carried the insertion scheme into other settings:

- A 1975 *Proceedings of the AMS* paper on inserting continuous functions between comparable real-valued functions cites Tong's paper as a primary reference and obtains four insertion theorems<sup>[2](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)</sup>.
- A 2007 *Acta Mathematica Hungarica* paper establishes a monotone lattice-valued version for functions taking values in ⊲-separable completely distributive lattices, together with a lattice-valued Urysohn lemma, giving new characterizations of monotonically normal spaces<sup>[8](https://link.springer.com/article/10.1007/s10474-007-6090-x)</sup>.
- A 2014 paper combines a σ-topological version of the Katětov–Tong theorem with a characterization of perfect σ-topologies to produce a Michael-type insertion theorem for measurable functions<sup>[9](https://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0128-3/)</sup>.
- A 2020 arXiv paper by Bezhanishvili, Morandi, and Olberding gives a new proof that first establishes the theorem for compact Hausdorff spaces and then extends it to all normal spaces via the [Stone–Čech compactification](https://www.edgepedia.org/stone-cech-compactification); in the compact case the underlying statement also implies a version of the [Stone–Weierstrass theorem](https://www.edgepedia.org/stone-weierstrass-theorem)<sup>[5](https://arxiv.org/abs/2001.08800)</sup>.
- A 2021 arXiv paper on insertion of continuous set-valued mappings shows that for normal spaces its set-valued insertion result is equivalent to the Katětov–Tong theorem, and characterizes τ-paracompact normal spaces by a set-valued insertion property<sup>[10](https://arxiv.org/abs/2109.12677)</sup>.

## Other work

One further publication by Tong is documented. "Non-existence of certain topological expansions", in *Annali di Matematica Pura ed Applicata* volume 86 (December 1970), proves that for every cardinal greater than 2^c there exists a completely regular space of that dispersion character which cannot be expanded to a normal space without changing the dispersion character; the paper states that this result answers a question raised by [Edwin Hewitt](https://www.edgepedia.org/edwin-hewitt)<sup>[4](https://doi.org/10.1007/bf02415707)</sup>.

## References

1. [G. Kubiak (1993). A strengthening of the Katětov–Tong insertion theorem. Commentationes Mathematicae Universitatis Carolinae.](https://cmuc.karlin.mff.cuni.cz/pdf/cmuc9302/kubiak.pdf)
2. [Proceedings of the American Mathematical Society, Vol. 49, No. 1 (1975), citing Hing Tong, Duke Math. J. 19 (1952), 289–292, MR 50265.](https://www.ams.org/journals/proc/1975-049-01/S0002-9939-1975-0362147-3/)
3. [Proceedings of the American Mathematical Society, Vol. 87, No. 3 (1983), citing M. Katětov, Fund. Math. 38 (1951), 85–91, MR 50264.](https://www.ams.org/journals/proc/1983-087-03/S0002-9939-1983-0684654-0/)
4. [Hing Tong (1970). Non-existence of certain topological expansions. Annali di Matematica Pura ed Applicata 86.](https://doi.org/10.1007/bf02415707)
5. [Bezhanishvili, Morandi, Olberding (2020). A new approach to the Katětov–Tong theorem. arXiv:2001.08800.](https://arxiv.org/abs/2001.08800)
6. [Good & Stares (2000). New proofs of classical insertion theorems. Commentationes Mathematicae Universitatis Carolinae 41.](https://www.dml.cz/bitstream/handle/10338.dmlcz/119148/CommentatMathUnivCarolRetro_41-2000-1_13.pdf)
7. [Czechoslovak Mathematical Journal 64 (2014), article discussing the Katětov–Tong insertion theorem.](https://www.dml.cz/bitstream/handle/10338.dmlcz/144054/CzechMathJ_64-2014-3_9.pdf)
8. [Monotone insertion of lattice-valued functions. Acta Mathematica Hungarica (2007).](https://link.springer.com/article/10.1007/s10474-007-6090-x)
9. [Inserting measurable functions precisely. Mathematica Bohemica (2014).](https://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0128-3/)
10. [Insertion of Continuous Set-Valued Mappings (2021). arXiv:2109.12677.](https://arxiv.org/abs/2109.12677)

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