hom·e·o·morphisms - 2025 PACE NSC

Question

A space is locally Euclidean if there exists one of these functions between a neighborhood of each point and some subset of R-n. For 10 points each:

[10h] Name these functions that preserve all topological properties. These continuous bijections between topological spaces have continuous inverses.

ANSWER: hom·e·o·morphisms [prompt on bicontinuous functions or topological isomorphisms by asking "what is the more common name?"; prompt on isomorphisms; reject "homomorphisms"]

[10m] Homeomorphisms preserve this property that is possessed by some subset of any neighborhood. A set with this property equals its interior and contains none of its boundary points.

ANSWER: openness

[10e] As bijections, homeomorphisms map precisely one element of this set to each element of this set's image. This set of inputs to a function is contrasted with the range.

ANSWER: domain

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Data

Team	Opponent	Part 1	Part 2	Part 3	Total
Adlai E. Stevenson A	Fremd	0	0	10	10
Alabama Cyber Tech	Wayzata	0	0	10	10
Barrington	Carmel B	0	10	10	20
Carl Sandburg	Georgetown Day	0	10	10	20
Centennial B	Uni School of Nashville B	0	0	10	10
Coronado	Walter Payton B	0	0	10	10
Detroit Catholic Central B	Thomas Jefferson C	0	0	10	10
Detroit Catholic Central C	Lincoln-Way East B	0	0	10	10
Eastchester	Detroit Catholic Central A	10	0	10	20
Glenbrook South	Northside College Prep	0	0	10	10
Great Valley	Seven Lakes	0	0	10	10
Hinsdale Central A	Adlai E. Stevenson B	0	0	10	10
Homewood-Flossmoor	Thomas Jefferson E	0	10	10	20
Kane County AT	North Quincy	0	10	10	20
Lambert B	Winston Churchill A	0	0	10	10
Lindbergh	Richard Montgomery	0	0	10	10
Milton	West Point	0	0	10	10
Montgomery Blair A	Johns Creek	0	10	10	20
Montgomery Blair B	Chattahoochee A	0	10	10	20
Northview	Belmont	10	10	10	30
Page	Yukon B	0	0	10	10
Park Tudor	Walter Payton A	0	0	10	10
Parkway West	Thomas Jefferson B	0	10	10	20
Quince Orchard	Innovation Academy B	0	0	10	10
Russell	Lincoln-Way East A	0	0	10	10
Saint Joseph	Herron	0	0	10	10
Saratoga	Classen SAS	0	0	10	10
Strake Jesuit B	Northmont	0	0	10	10
Team 1	Team 2	0	0	10	10
Thomas Jefferson A	Innovation Academy A	0	0	10	10
Thomas Jefferson D	Chattahoochee B	0	0	10	10
Uni School of Nashville A	Centennial A	0	0	10	10
Uni of Illinois Laboratory	Lambert A	0	0	10	10
Virginia	Winston Churchill B	0	0	10	10
Westwood	Grosse Pointe South	0	10	10	20
White Pigeon	Lee's Summit	0	0	10	10

Summary

Tournament	Edition	Exact Match?	Heard	PPB	Easy %	Medium %	Hard %
2025 PACE NSC	06/07/2025	Y	36	13.06	100%	25%	6%