compactness - 2026 ACF Regionals

Question

Spaces with this property are conceptually the topological analogue of a finite set in set theory. For 10 points each:

[10m] Name this ubiquitously discussed property in analysis and topology held by subsets of Rn (“R-N”) that are closed and bounded.

ANSWER: compactness [or compact spaces or compact sets]

[10h] This theorem states that an equicontinuous and bounded real-valued function on a compact set converges uniformly. It is often proven by taking a countable dense subset of a space, then using the Bolzano–Weierstrass theorem, then diagonalizing to construct a convergent subsequence.

ANSWER: Arzela–Ascoli theorem [or Ascoli–Arzela theorem]

[10e] Uniform convergence holds if the difference between terms becomes arbitrarily small for some N that depends only on a quantity denoted by this letter. Continuity and convergence are formalized by a definition named for this letter and delta.

ANSWER: epsilon [or epsilon–delta definition]

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Summary


California (North)	Main Site	Y	3	26.67	100%	100%	67%
Canada	Main Site	Y	10	10.00	60%	30%	10%
Florida	Main Site	Y	3	13.33	67%	67%	0%
Great Lakes	Main Site	Y	5	10.00	80%	20%	0%
Midwest	Main Site	Y	8	16.25	75%	75%	13%
Northeast	Main Site	Y	5	16.00	80%	60%	20%
Overflow	Main Site	Y	4	12.50	100%	25%	0%
South Central	Main Site	Y	2	10.00	50%	50%	0%
Southeast	Main Site	Y	7	12.86	86%	29%	14%
UK	Main Site	Y	12	20.00	92%	83%	25%
Upper Mid-Atlantic	Main Site	Y	8	15.00	75%	63%	13%
Upstate NY	Main Site	Y	4	17.50	100%	75%	0%

Data


Berkeley A	Stanford B	10	10	10	30
Berkeley C	Berkeley B	10	0	10	20
Stanford A	Berkeley D	10	10	10	30
Brock	McGill A	0	0	0	0
Carleton University	Toronto G	0	0	10	10
Guelph	Western A	0	0	0	0
McGill B	Waterloo B	0	0	10	10
Western C	McMaster A	0	0	0	0
Ottawa	McMaster B	10	0	10	20
Toronto Metropolitan	Queen's	0	0	0	0
Toronto E	Toronto B	10	10	10	30
UBC B	Toronto C	10	0	10	20
Western B	Simon Fraser	0	0	10	10
Central Florida A	Central Florida B	10	0	10	20
CWRU A	Ohio State A	0	0	10	10
CWRU B	Michigan A	0	0	0	0
Michigan B	Ohio State B	10	0	10	20
Michigan C	Michigan State	0	0	10	10
Pittsburgh	Ohio State C	0	0	10	10
Florida A	Central Florida C	10	0	10	20
Chicago A	Purdue B	10	10	10	30
Purdue A	Chicago B	10	0	10	20
Chicago C	Northwestern B	10	0	10	20
Illinois A	Chicago D	10	0	10	20
Illinois C	Illinois B	10	0	10	20
Notre Dame B	Indiana C	0	0	0	0
Indiana A	Northwestern C	10	0	10	20
Indiana B	Notre Dame A	0	0	0	0
Florida C	Florida B	0	0	0	0
Brown	Harvard B	10	10	10	30
Dartmouth A	Bowdoin A	0	0	10	10
Brandeis A	Yale	0	0	0	0
Georgia Tech A	Claremont	0	0	10	10
Iowa State	Washington B	0	0	10	10
Missouri A	Carleton College	10	0	10	20
Washington A	Missouri B	0	0	10	10
Texas A	Murray State A	10	0	10	20
Texas B	Texas A&M A	0	0	0	0
Harvard A	Dartmouth B	10	0	10	20
Georgia Tech B	Alabama A	0	0	10	10
Bruin	Georgia Tech D	0	10	10	20
Georgia Tech C	Alabama B	10	0	10	20
Tennessee B	Mississipi State	0	0	0	0
Louisville B	Sewanee	0	0	10	10
Tennessee A	WashU A	10	0	10	20
Vanderbilt B	Kentucky	0	0	10	10
MIT	Southern Connecticut State	10	0	10	20
Oxford A	Bristol	10	10	10	30
Imperial C	Cambridge C	10	0	10	20
Cardiff	Southampton A	0	0	0	0
Oxford B	City St. George's	10	0	10	20
Imperial A	Durham	10	10	10	30
Edinburgh A	Oxford C	10	0	10	20
LSE	Edinburgh B	10	0	10	20
Manchester	Montpellier	10	0	10	20
Imperial B	Southampton B	10	10	10	30
Cambridge B	UCL	0	0	10	10
Warwick A	Birmingham	10	0	10	20
Cambridge A	Warwick B	10	0	10	20
Maryland A	Columbia C	10	0	10	20
Columbia A	Vassar	10	0	10	20
Maryland B	John Jay	0	0	10	10
Columbia B	Johns Hopkins A	10	10	0	20
Swarthmore	NYU C	10	0	10	20
Cornell A	Binghamton B	0	0	10	10
Cornell C	Binghamton A	10	0	10	20
Cornell B	RIT	10	0	10	20
Penn State	Rochester	10	0	10	20
Maryland C	Georgetown	0	0	10	10
NYU A	Johns Hopkins B	10	0	10	20
Catholic	NYU B	0	0	0	0