Topology: Concepts & Examples

Question 1

topological space

Answer 1

-0 and X are open
-closed under arbitrary unions
-closed under finite intersections

Question 2

topology

Answer 2

set of open subsets of X

Question 3

closed

Answer 3

complement is open

Question 4

neighbourhood

Answer 4

subset A such that there exists U in A

Question 5

discrete topology

Answer 5

all subsets are open

Question 6

indiscrete topology

Answer 6

only trivial subsets are open

Question 7

continuous map

Answer 7

inverse image of open sets is open
inverse image of closed sets is closed

Question 8

homeomorphism

Answer 8

continuous bijections with continuous inverse

Question 9

open map

Answer 9

f(U) is open for all U

Question 10

base

Answer 10

each non-empty open set in X is a union of elements of B

Question 11

subbase

Answer 11

the collection of intersections of S forms a base

Question 12

induced topology

Answer 12

f continuous iff i o f continuous

Question 13

product topology

Answer 13

f continuous iff pri o f continuous

Question 14

quotient topology

Answer 14

f continuous iff f o π continuous

Question 15

topological group

Answer 15

group and ab and a^-1 are both continuous

Question 16

general linear group

Answer 16

invertible matrices

Question 17

orthogonal group

Answer 17

AA^T=I

Question 18

special orthogonal group

Answer 18

det=1

Question 19

compact-open topology

Answer 19

sublease is {compact to open continuous maps}