Topology
Prove or disprove ( with a counterexample)
(a) The continuous image of a Hausdorff space is Hausdorff.
(b) The continuous image of a connected space is
connected.
Topology question:
Show that a function f : ℝ → ℝ is continuous in the ε − δ
definition of continuity if and only if, for every x ∈ ℝ and every
open set U containing f(x), there exists a neighborhood V of x such
that f(V) ⊂ U.
Show that any open subset of R (w. standard topology) is a
countable union of open intervals. Please explain how to do, I only
understand why it is true.
What is required to fully prove this. What definitions should I be
using.
Please justify below questions (its a topology question) with
valid arguments:
1. Show that an infinite set is not finite.
2. Is there an infinite set which is not countably in
finite?
Thanks in advance.
Topology
(a) Prove that the interval [0,1] with the subspace topology is
connected from basic principles.
(b) Prove that the interval [0,1] with the subspace topology is
compact from basic principles.
Calculate the relative (sub-space) topology with respect to the
usual (metric) topology in R (the set of real numbers), for the
following sub-sets of R:
X = Z, where Z represents the set of integers
Y = {0} U {1 / n | n is an integer such that n> 0}
Calculate (establish who are) the closed (relative) sets for the
X and Y sub-spaces defined above.
Is {0} open relative to X?
Is {0} open relative to Y?