Martin Sleziak's room

« first day (2517 days earlier) ← previous day next day → last day (912 days later) »

8:18 AM

In [Tka] the author writes: "Every topological space $X$ can be represented as an open continuous image of a completely regular submetrizable space $Y$ (in other words, $Y$ admits a continuous one-to-one mapping onto a metrizable space) — the corresponding construction is given on p. 331 of [Eng]"...

A: Open images of submetrizable spaces

It seems quite likely that it was intended to be page 331 of the Engelking's book published in 1977 in PWN, Warszava1 - rather then the later edition by Heldermann (revised and completed edition, 1989). I will quote in full exercise 4.2.D (page 331 in the older edition, page 264 in the newer edit...

Although this seems to be very likely the place in Engelking's book that the author had in mind, I do not immediately see how this construction works for submetrizable spaces. (Hopefully, some people around here are familiar enough with submetrizable spaces to give some advice on that.) — Martin Sleziak yesterday

The answer is contained in Junnila's paper. From the scanned page to which you have: "every topological space is the continuous image of a sigma-discrete stratifiable space under an open map" (every paracompact space with a $G_\delta$-diagonal is submetrisable). — Tyrone 1 hour ago

12 hours later…

Feeds

8:14 PM

Q: Planned maintenance scheduled for Oct 30+31, 2021

Due to ongoing planned network upgrades, all public Stack Exchange properties (stackoverflow.com, stackexchange.com, and so on) will be in read-only mode during two maintenance windows: Saturday October 30 and Sunday, October 31, 2021. While in read-only mode, the websites will be accessible but ...