Urysohn 39;s Lemma

[Sandi Loisel]

Urysohns lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalised by (and usually used in the proof of) the Tietze extension theorem.

Two subsets A \displaystyle A and B \displaystyle B of a topological space X \displaystyle X are said to be separated by neighbourhoods if there are neighbourhoods U \displaystyle U of A \displaystyle A and V \displaystyle V of B \displaystyle B that are disjoint. In particular A \displaystyle A and B \displaystyle B are necessarily disjoint.

It follows that if two subsets A \displaystyle A and B \displaystyle B are separated by a function then so are their closures. Also it follows that if two subsets A \displaystyle A and B \displaystyle B are separated by a function then A \displaystyle A and B \displaystyle B are separated by neighbourhoods.

A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T1 spaces are Tychonoff.

Remarkably, the result is true in greater generality: namely, for Hausdorff, second-countable smooth manifolds. It is the fundamental result upon which smooth partitions of unity are constructed on smooth manifolds as above. A proof may be found in lemma 1.3.2 on pages 10-11 of Peter Petersen's "Manifold Theory"

This document discusses Urysohn's lemma, which states that a topological space is normal if any two disjoint closed subsets can be separated by a continuous function. It provides background on mathematician Pavel Urysohn, defines key terms like normal space, and outlines the proof of Urysohn's lemma, which constructs a continuous function separating two disjoint closed sets using dyadic fractions. Applications of the lemma include formulating other topological properties and solving extension theorems.Read less

I covered the material in the "Brief review of sets and function" (more or less) through the section called "Functions". On Thursday I shall talk about countability from the same notes and then continue with chapter 2 of the textbook.

I first finished the "Brief review of sets and functions" by going through the material on countability, and then turned to section 12 in the book. Here I defined topologies and showed some examples (in addition to those in the book, I proved that the open sets in a metric space form a topology). I then turned to section 13 where I introduced the notion of a basis for a topology and proved that all bases generate topologies. On Friday I shall start with Lemma 13.1.

I first completed section 13 by proving Lemma 13.1 and showing how a subbasis can be used to define a topology. I then defined neighborhoods and convergence of sequences, and tried to use these concepts to give an intuitive idea of how topologies describe "how close together" points are. I then rather quickly covered $$14, 15 and 16. Tuesday I shall talk on the important $17.

I first introduced closed sets (after some motivation from metric spaces) and proved Theorem 17.1. I then introduced the closure of a set and proved Theorem 17.5. Then we turned to limit points and proved Theorem 17.6 and its corollary. Next I defined Hausdorff spaces and proved Theorems 17.8 and 17.10. I ended the lecture by trying to motivate the abstract definition of continuous functions in $18.

Today I covered most of $18 on continuous functions. I first defined continuity and then proved Theorem 18.1 (which I broke down into three separate propositions). I said a a few words about homeomorphisms and imbeddings, and then pointed out that to prove that a function is continuous, it suffices to show that the inverse image of all sets in a basis or a subbasis are open. I proved Theorem 18.4 as an illustration of this technique. Next time I shall probably prove the pasting lemma (Theorem 18.3) before I move on to $19.

Today I covered most of $22 on quotient topologies. I first tried to motivate the theory by introducing the idea of a topology induced by a mapping or by an equivalence relation. Next I introduced quotient maps, and proved that they can be characterized as the continuous, surjective functions which map open, saturated sets to saturated sets. I then introduced quotient topologies and proved that they really are topologies. I then turned to examples and spent some time on the construction of a quotient topology on the torus (example 5 in the book). Finally, I proved the first part of the first part of Theorem 22.1. Next time I shall prove Theorem 22.2 and 22.3, and then start $23.

I finished the material on quotient topologies by first proving that the composition of two quotient maps is a quotient map and that a bijection is a quotient map if and only if it is a homeomorphism. I then used these results to prove Theorem 22.2 and its corollary (only the first part). I then started Chapter 3 by talking informally about connectedness and path connectedness before I defined connected spaces and proved Lemma 23.1, Lemma 23.2 and Theorem 23.3.

I went through 26 on compactness with the exception of Theorem 26.9 (which I didn't have time for, but which I shall return to next time). In addition to what is in the book, I sketched the proof that a closed, bounded interval in R is compact (this is basically the proof of Theorem 27.1, which I shall not return to).

I started by saying a few words about Tychonov's Theorem which says that the product of an arbitrary family of compact sets is compact (in the product topology). This theorem is not on the curriculum (but in the book, see 37), but important to know about.

I then introduced the finite intersection property and wrote down Theorem 26.9, leaving the proof as an exercise. I went rather quickly through 27 as most of it is well-known from MAT1300, just picking up the extreme value theorem (27.4) and the Lebesgue number lemma (27.5). I then introduced limit point compactness and sequential compactness, and proved Theorems 28.1 and Theorem 28.2.

I started by defining regular and normal spaces, and proved Lemma 31.1. Then I stated Theorem 31.2 and proved part b), before turning to Theorem 32.1 and its proof. I shall continue next time with theorems 32.2 and 32.3, and will then probably turn to section 33.

I defined completely regular spaces and wrote down Theorem 33.2 (without proof) before I turned to Urysohn's Metrization Theorem (Theorem 34.1). I described the theorem and gave an outline of its proof before turning back to 20 and the proof of theorem 20.5 (metrizability of R^N). I then reurned to 34 where I first proved "Step 1" as a separate lemma, and then finished the proof of the Urysohn Metrization Theorem (the "first version" in the book).

I first proved Ascoli's theorem (version 45.4), and then started 46 where I introduced the topology of pointwise convergence and sketched the proof of Theorem 46.1. I then turned to the topology of uniform convergence on compacts and theorem 46.2.

I dropped the bit on compactly generated spaces (lemma 46.3-theorem 46.7) and went straight to the compact-open topology where I proved Theorem 4.6.8, Theorem 4.6.9 and Theorem 4.6.11). Next time I start chapter 9.

Finished 52 and started 53. In the latter I put the emphasis of an intuitive understanding of covering spaces, and I spent most of the time on the covering spaces of the circle (in the form of a helix) and the torus (as in Example 4). I finished with the formal definitions of "evenly covered" and "covering space".

I started by proving that the n-sphere is simply connected for n>1 and then turned to 60 where I proved all the results (cheating a bit in the proof of Theorem 60.3 by not checking in detail that p is really a covering map). This means that we have completed the syllabus and will use the last two lectures for review.

I started the review by recalling the basic notions of a topology, open and closed sets, and continuous functions. I then mentioned how we can build topologies from bases and subbases, and recalled subspace topologies, product topologies, quotient topologies and metric topologies, and said a few words about how we in practice prove that sets are open or closed and functions are continuous (here I mentioned pointwise continuity and the pasting lemma). I then went on to talk on connedtedness: its definition, the definition of components, that products and closures of connected sets are continuous, and that unions of connected sets are connected when they have at least one point in common. I then turned to compactness where I mentioned the definition, the alternative definition in terms of closed sets ("families with the finite intersection property") and the fact that products of compact sets are compact. Finally, I briefly mentioned the countability and separation axioms, Urysohn's lemma and Urysohn's metrization theorem.

I continued the review by briefly mentioning the completness of C(X;Y) (Theorem 43.6) and Ascoli's theorem before turning to Chapter 9 and algebraic topology. I defined homotopies and path homotopies and introduced the fundamental group, mentioning that in path connected spaces, this group is independent of the base point. I then introduced covering spaces, and recalled the ways in which paths and homotopies can be uniquely lifted to the covering space (including the fact that when the covering space is simply connected, there is a bijection between the fundamental group and the fiber of the base point, Theorem 54.4). I then described how we can "compare" fundamental groups using retracts, deformation retracts, homotopy equivalences and theorems 59.1 and 60.1. As we had some extra time I covered Example 2 on page 362 as an example of deformation retracts,

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