How did submetrizability come into existence ? What is submetrizability of a topological space used for?

Question

Can anybody tell me how did submetrizability come into existence, and what is its use in topology ? Any examples to make me understand ?

Answer 1

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.

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Added based on comments: different guises under which we can meet this property:

TFAE:

1. $(X, \tau)$ is submetrisable.

2. 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 $(X,\tau) \times (X,\tau)$.

3. 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.

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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)

Answer 2

I want to chip in to the nice answer by Henno Brandsma, with this small fact:

If $(X,τ)$ is submetrizable space, then any compact $K ⊆ X$ is metrizable.

Proof. Let $τ_d$ be a topology on $X$ inudced by a metric. Any compact $K\subset X$ is submetrizable as a subset of submetrizable space. We have $τ_d|K \subseteq τ|K$. Thus $K$ is compact in $τ_d|K$ as well (1). But $τ_d|K$, as a metric space, is Hausdorff and Hausdorff topology on a compact space is the largest possible (2). Thus $τ_d|K = τ|K$.