Can anybody tell me how did submetrizability come into existence, and what is its use in topology ? Any examples to make me understand ?
How did submetrizability come into existence ? submetrizability of a topological space used for
general-topologymetrizabilityterminology
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In the old usage, as well as contemporary usage in set theory, one may consider a function without specifying a particular codomain or target set. (The insistence that a function come along with a particular codomain is a comparatively recent innovation, probably arising in Bourbaki.)
That is, if one understands a function merely to be a set of ordered pairs satisfying the function property (that each input is associated to one output), or as a rule associating to every object in a domain an output value, then it is true to say that a function is one-to-one if and only if it is a bijection from its domain to its range. Thus, injective functions really are one-to-one in the sense that you want.
Of course, this one-to-one terminology was long established by the time Bourbaki wanted to insist that functions come along with a specified co-domain, giving the definition of function as a triple consisting of domain, codomain and set of ordered pairs. The fact that in this context the concept of one-to-one doesn't tell the whole story may be part of the reason that they introducted the injective, surjective, bijective terminology.
But meanwhile, a function is one-to-one if and only if it provides a one-to-one correspondence between its domain and its range. This is perfectly logical, and seems to be the explanation that you are seeking. I would think that the one-to-one terminology begins to seem illogical only when one also insists on attaching to the function a target set or codomain that is not the same as its range, which is, after all, a somewhat illogical thing to do.
I like to think of topological spaces as defining "semidecidable properties". Let me explain.
Imagine I have an object that I think weighs about one kilogram. Suppose that, as a matter of fact, the object weighs less than one kilogram. Then I can, using a sufficiently accurate scale, determine that the object weighs less than one kilogram. Even if the object weighs, say, 0.9999996 kilograms, all I need to do is find a scale that's accurate to within, say, 0.0000002 kilograms, and that scale will be able to tell me that the object weighs less than one kilogram.
This means that "weighing less than one kilogram" is a semidecidable property: if an object has the property, then I can determine that it has the property.
Suppose, on the other hand, that the object actually weighs exactly one kilogram. There's no way I can measure the object and determine that it weighs exactly one kilogram, because no matter how precisely I measure it, it's still possible that there's some amount of error which I haven't discovered yet. So "weighing exactly one kilogram" is not a semidecidable property.
What does this have to do with topological spaces? Well, an open set in a topological space corresponds to a semidecidable property of that space. This is why in the topological space of real numbers, the set $\{x : x \in \mathbb{R}, x < 1\}$ is an open set, but the set $\{x : x \in \mathbb{R}, x = 1\}$ is not.
So, consider the "topological space" $X = \{a, b, c\}$ with open sets $\emptyset$, $\{a, b\}$, $\{b, c\}$, and $\{a, b, c\}$. In this "topological space", you are asserting that
- (since $\{a, b\}$ is open) if you have a point which is either $a$ or $b$, then it is possible to measure it and determine that it is either $a$ or $b$ (though it is not necessarily possible to determine which one it is);
- (since $\{b, c\}$ is open) if you have a point which is either $b$ or $c$, then it is possible to determine that it is either $b$ or $c$; but
- (since $\{b\}$ is not open) if you have the point $b$, it is not possible to determine that it is $b$.
However, these assertions contradict each other. Suppose that you have the point $b$. Because of the first bullet point, there is some measurement you can make which will tell you that the point is either $a$ or $b$. And because of the second bullet point, there is another measurement you can make which will tell you that the point is either $b$ or $c$. If you simply make both of these two measurements, then you will have successfully determined that the point is (either $a$ or $b$, and either $b$ or $c$)—in other words, that the point is $b$. But the third bullet point asserts that this is impossible!
For more explanation of this idea, see these two answers:
- https://math.stackexchange.com/a/31946/13524 (Qiaochu Yuan's answer to "What concept does an open set axiomatise?")
- https://mathoverflow.net/a/19531/5736 (community wiki answer to "Why is a topology made up of 'open' sets?")
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Best Answer
There is a whole subbranch of general topology that came out of finding and proving metrisation theorems, and a lot of properties have been found that are so-called "Generalised Metric Spaces", that have nice properties and are "close to" metrisable in some sense. Monotonically normal spaces are such spaces (I recently talked about them here, in connection with the Michael line), developable spaces (introduced by Bing), spaces with a $G_\delta$ diagonal, $M$-spaces, $p$-spaces, etc. See Gruenhage's survey paper in the Handbook of Set theoretic Topology.
Many examples of such spaces (non-metrisable ones) do have the property, it was observed, that they have a natural weaker metrisable topology on the same set, e.g. the Sorgenfey line, the Michael line (which have the Euclidean topology as a weaker topology (a subtopology). So it is logical to start studying that feature as a new property in its own right, and call it submetrisable (British spelling). We can then prove interesting equivalent formulations of it (see Gruenhage's article) which make it look like other properties and prove some implications (like: submetrisable implies having a $G_\delta$ diagonal) so that it fits in a network of other properties. I cou;ldn't find who first introduced the property or its name (looks a bit like a property that Arhangel'skij would introduce to me) but it does fit into a wider family of properties.
Added based on comments: different guises under which we can meet this property:
TFAE:
$(X, \tau)$ is submetrisable.
There is a metric $d: X \times X \to \Bbb R$ (so a function obeying the metric space axioms) on $X$ that is continuous on $(X,\tau) \times (X,\tau)$.
There is a continuous injection $f(X,\tau) \to (Y,d)$, where $(Y,d)$ is a metric space (of course having the metric topology $\tau_d)$.
$1 \to 2$: Let $\tau' \subseteq \tau$ be a topology on $X$ that is induced by some metric $d$ on $X$. Then $d$ is continuous as a map on $(X,\tau') \times (X, \tau')$, this is classical, and so the same holds for the stronger topology on $X^2$, $(X,\tau) \times (X,\tau)$.
$2 \to 3$: Let $Y=(X,d)$ and $f(x)=x$. $Y$ is by definition metric and $f$ an injection and $d$-open balls on $X$ are $\tau$-open as $B_d(x,r) = d_x^{-1}[(-\infty,r)]$, where $d_x: (X,\tau) \to \Bbb R, d_x(y)= d(x,y)$ is continuous as $d$ is. It follows that $f$ is continuous, as required.
$3 \to 1$: Given $f:(X, \tau) \to (Y,d)$ continuous and injective, define $\tau'$, a topology on $X$, by $\{f^{-1}[O]\mid O \in \tau_d\}$ and note that $\tau' \subseteq \tau$ and $(X,\tau') \simeq (f[X], d)$ so $X$ is submetrisable.
These are studied for being interesting, they have no "use" whatsoever. What's the use of an $L$-space? It's just expanding our knowledge of what's possible in topology.. I don't think it occurs as a condition in a theorem that is widely applied in analysis or elsewhere e.g. It's just one of the hundreds of properties that topologists have introduced for the sake of exploring. One interesting theorem (IMO): Every paracompact space with a $G_\delta$ diagonal is submetrisable. (If we replace paracompact by compact we get a metrisable space, then follows)