The space Q is not connected. In fact, for any irrational x∈R, the set [−x,x]∩Q=(−x,x)∩Q is both open and closed in the subspace topology on Q.
Adv.
If A is connected and B is a set such that A⊆B⊆¯A then B is also connected.
Show that the complement of any countable subset C in R2 is path-connected.
Show that the continuous image of a path-connected space is path-connected.
For open subset U of Rn , show that U is connected if and only if U is path-connected.
Let X be a compact space with a nested sequence {Cn} of non-empty closed subsets: C1⊇C2⊇C3⋯ . Show that the intersection ∩nCn is non-empty.
A topological space X is compact if and only if for every collection C with finite intersection property, ∩C∈CC is non-empty.
Let X be a first countable space and let A be a subset of X. Then the following are true:
(1) (Sequence Lemma) A point x∈¯A if and only if there is a sequence of points of A converging to x.
(2) (Continuity Versus Sequential Continuity) Let f:X→Y . Then f is continuous if and only if f is sequentially continuous.
A metric space (X,d) is second countable if any one of the following holds true:
(1) X has a countable dense subset,
(2) X is compact.
Show that Rl is not metrizable.
If X is second countable then every open covering of X contains a countable subcover.
All Questions
Physics
Chemistry
Mathematics
English
Organic Chemistry
Inorganic Chemistry
Physical Chemistry
Algebra
Geometry