- #1

Gear300

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This did not come from a topology book, but we were asked to prove Urysohn's Lemma. We are familiar with standard proofs for this, which are all likely simpler to exhibit than our attempt here, but we were just curious about where our method here went wrong.

At a glance, the lemma essentially states that all normal spaces bear a topology at least as strong as the metric space R

^{1}∩ [0,1]. Since preimages preserve many set operations, it seems a good deal of the problem can be handled just by thinking in terms of [0,1].

Let T be the mother set, and let F and G bet the closed sets in question. The idea was to inductively tackle the problem.

Stage 1: If F and G are our closed sets, then we can let F be a subset of the preimage of 0 and G be a subset of the preimage of 1. By normality, we can arbitrate that the closed set constituting the complement of their disjoint open covers is exactly the preimage of ½, while the open cover of F is the preimage of [0,½) and the open cover of G is the preimage of (½,0]. So at this point, we have identified the open sets

S = { ƒ

^{-1}( [0,½) ) , ƒ

^{-1}( (½,1] ) },

and the closed sets

S = { F ⊆ ƒ

^{-1}(0) , G ⊆ ƒ

^{-1}(1) , ƒ

^{-1}( [0,½] ) , ƒ

^{-1}( [½,1] ) , ƒ

^{-1}(½) }.

It is important to ensure that all set operations between the preimages make sense for the sake of consistency, and they intuitively seem to.

Stage 2: We can then take the complement of the open cover of F to induce another closed set, which should be the preimage of (½,1]. We can call this set X. By normality, F and X bear disjoint open covers, so we can let the closed set constituting their complement to be the preimage of ¼. Similarly, we can do the same thing with G and the complement of its open cover to attain the preimage of ¾. So at this point, we should have been able to identify the open sets,

S = { ƒ

^{-1}( [0,½) ) , ƒ

^{-1}( (½,1] ) , ƒ

^{-1}( [0,¼) ) , ƒ

^{-1}( (¼, ¾) ) , ƒ

^{-1}( (¾,1] ) },

and

S = { F , G , ƒ

^{-1}(¼) , ƒ

^{-1}(½) , ƒ

^{-1}(¾) , ƒ

^{-1}( [0,½] ) , ƒ

^{-1}( [½,1] ) , ƒ

^{-1}( [0,¼] ) , ƒ

^{-1}( [¼,¾] ) , ƒ

^{-1}( [¾,1] ) }.

And again, it is important to ensure that all set operations between the preimages make sense...and they seem to.

So the general idea is that we can continue identifying new open and closed sets by progressively inducting on our current set of open and closed sets via basic set operations and a property of normal spaces. The main points are that

1. At every nth stage, we achieve the (closed) preimages of the dyadic rationals in [0,1] that are a multiple of 1/2

^{n}.

2. We progressively achieve smaller open covers around each dyadic rational as the stages progress. The radii of the open covers decrease by ½ with each stage ever since the stage of their inception.

e.g. The closed set ƒ

^{-1}(1/2) has open cover ƒ

^{-1}( (¼,¾) ) at stage 2, and at stage 3, it also gains the open cover ƒ

^{-1}( (3/8 , 5/8) ).

So what we altogether accomplish, if this is right, is an open preimage of each open sphere in a local neighborhood base surrounding each dyadic rational in the interval [0,1]. Each such local neighborhood base is a countable set of open spheres of radii 1/2

^{n}. Since the set of dyadic rationals are dense in [0,1], a countable union across this countable set of local neighborhood bases returns a countable base for the topology in R

^{1}∩ [0,1]. By theorem, given ƒ : U → V, if for every open set A ⊆ V, its preimage ƒ

^{-1}(A) is open, then ƒ is continuous. So the mapping we have constructed should be continuous, since the preimage of any (open) set in our just-constructed base is open.

The odd part about this is that it seems we could have let F = ƒ

^{-1}(0) and G = ƒ

^{-1}(1) right at the beginning. Then if all of our arguments are consistent, our function should precisely separate the two closed sets, which is hallmark of a perfectly normal space and not normal spaces in general. The added perk is that as our open covers converge around each dyadic rational, their preimages should converge to the closed preimage of the dyadic rational. This matches the idea that a space is perfectly normal iff every closed set is a G

_{δ}set.

In any case, we came up with this on the spot without double-checking. We may be blunt in asking, but where exactly is our argument wrong? Or at the very least, what are the various differences between normal and perfectly normal spaces?

Also, how does Urysohn's Lemma apply to finite normal spaces?