« first day (278 days earlier)      last day (2081 days later) » 

user131753
03:26
in Logic, yesterday, by famesyasd
1) An open cover of A is a collection of open sets in the underlying topology whose union includes all of A.
2) An open cover of A is a collection of open balls in the underlying topology whose union includes all of A.
Does one require the axiom of choice to prove that two definitions are equivalent?
user131753
@famesyad: Not sure what you mean by equivalence. However, if you mean the following:
user131753
"Let $(X,d)$ be a metric space and $A\subseteq X$. An open cover $\mathscr{U}\subseteq \mathcal{P}(X)$ of $A$ induces an open cover $\mathscr{V}$ of $A$ by open balls and vice versa."
user131753
Then you don't need $\mathsf{AC}$ to prove this.
user131753
03:41
Here is a sketch of an argument: Let $\mathscr{U}$ be an open cover of $A$ and $U\in\mathscr{U}$. Then since $U$ is open, for all $x\in U$ there exists $\varepsilon_x>0$ such that $B_U(x,\varepsilon_x)\subseteq U$. Show that $$\displaystyle\bigcup_{x\in U}B_U(x,\varepsilon_x)= U$$Then define, $$\mathscr{V}:=\{B_U(x,\varepsilon_x):U\in \mathscr{U}~\text{and}~x\in U\}$$
user131753
The reverse part is obvious.
chytrik
chytrik
07:20
← 7 messages moved from mempool
Feeds
Feeds
07:47
BBC Sport - Rugby Union: New Zealand's Brodie Retallick to take break from All Blacks and play in Japan
posted on June 12, 2019

Former World Rugby player of the year Brodie Retallick signs a new deal with New Zealand Rugby that will see him take a break from the All Blacks.

John Rennie
John Rennie
08:39
In other news, it has stopped raining in Chester after 48 hours of continuous rain.
Apparently this was due to a European weather system that pushed in the UK from the south east.
After Brexit we will be able to stop continental weather systems from entering the UK unless they have a visa.
The government assures me of this so it must be true.
← 4 messages moved from Problem Solving Strategies
SmokeDetector
SmokeDetector
09:22
[ SmokeDetector | MS ] Pattern-matching website in body, potentially bad keyword in body (98): Vapi Escorts Services by Vapi Escorts on ell.SE
Feeds
Feeds
10:06
BBC Sport - Rugby Union: Gatland named British and Irish Lions coach for 2021 South Africa tour
posted on June 12, 2019

Warren Gatland is named British and Irish Lions coach for a record-equalling third time and will lead the tour of South Africa in 2021.

Feeds
Feeds
10:26
BBC Sport - Rugby Union: Jed McCrory: Worcester Warriors director leaves Premiership club
posted on June 12, 2019

Worcester Warriors co-owner Jed McCrory leaves the Premiership club just eight-and-half months after leading a takeover.

M.A.R. ಠ_ಠ
M.A.R. ಠ_ಠ
← 1 message moved from Language Overflow
 
2 hours later…
Feeds
Feeds
12:26
BBC Sport - Rugby Union: Ronan O'Gara to move from Crusaders role to take La Rochelle head coach post
posted on June 12, 2019

Former Ireland and British and Irish Lions fly-half Ronan O'Gara is to become head coach at Top 14 club La Rochelle.

Feeds
Feeds
12:46
BBC Sport - Rugby Union: Scotland players 'must up intensity' for British & Irish Lions selection
posted on June 12, 2019

Scotland players need to maintain the levels shown in this year's dramatic Twickenham fight back if they are to have a chance of British and Irish Lions selection, says Scott Hastings.

 
1 hour later…
Feeds
Feeds
14:06
BBC Sport - Rugby Union: Quins' Catrakilis to return to South Africa
posted on June 12, 2019

Harlequins fly-half Demetri Catrakilis returns to South Africa to sign for Pro 14 side Southern Kings.

BBC Sport - Rugby Union: Jon Welsh: Scotland prop agrees new two-year Newcastle Falcons deal
posted on June 12, 2019

Scotland prop Jon Welsh signs a new two-year contract with Newcastle.

 
3 hours later…
Feeds
Feeds
17:26
BBC Sport - Rugby Union: Warren Gatland: British and Irish Lions coach ‘will not be coaching England’ after tour
posted on June 12, 2019

Confirmed British and Irish Lions head coach Warren Gatland says he "will not be coaching England" after the Lions' 2021 tour to South Africa.

BBC Sport - Rugby Union: Gatland rules out future England job after being confirmed as Lions coach
posted on June 12, 2019

Warren Gatland rules out succeeding Eddie Jones as England coach and plans to return to his native New Zealand after leading the British and Irish Lions in 2021.

 
3 hours later…
SEJPM
SEJPM
19:56
← 4 messages moved from The Side Channel
← 13 messages moved from The Side Channel
← 6 messages moved from The Side Channel
← 2 messages moved from The Side Channel
← 1 message moved from The Side Channel
← 26 messages moved from The Side Channel
← 6 messages moved from The Side Channel
← 6 messages moved from Cryptography-Politics
SmokeDetector
SmokeDetector
21:01
[ SmokeDetector | MS ] Few unique characters in answer, repeating words in answer (173): Need answer to an "if" question by Ahmed on ell.SE
 
3 hours later…
M.A.R. ಠ_ಠ
M.A.R. ಠ_ಠ
23:52
← 1 message moved from Language Overflow

« first day (278 days earlier)      last day (2081 days later) »