Question

Prove the conjecture or provide a counterexample:

Let U ∈ in the usual topology, and let F be a finite set. Then (U−F) ∈ in the usual topology.

Answer #1

Prove or provide a counterexample
Let f:R→R be a function. If f is T_U−T_C continuous, then f is
T_C−T_U continuous.
T_U is the usual topology and T_C is the open half-line
topology

Prove that in R^n with the usual topology, if a set is
closed and bounded then it is compact.

Prove that, as a subspace of R with its usual topology, Z has the
discrete topology.

Prove or provide a counterexample
Let (X,T) be a topological space, and let A⊆X. Then A is dense
iff Ext(A) =∅.

Prove or provide a counterexample
If A is a nonempty countable set, then A is closed in T_H.

Prove that a closed set in the Zariski topology on K1 is either
the empty set, a finite collection of points, or K1 itself.

Let X be a topological space with topology T = P(X). Prove that
X is finite if and only if X is compact. (Note: You may assume you
proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2
and simply reference this. Hint: Ô⇒ follows from the fact that if X
is finite, T is also finite (why?). Therefore every open cover is
already finite. For the reverse direction, consider the
contrapositive. Suppose X...

Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture: There does not exist a pair of integers m and n such
that m^2 - 4n = 2.

True or False: Any finite set of real numbers is complete.
Either prove or provide a counterexample.

Let the set N of natural numbers be endowed with the cofinite
topology (in which a set is open if and only if it is empty or its
complement is finite). (a) Is N connected? Justify your answer. (b)
Is N compact? Justify your answer. (c) Explain why the function f :
N → N, n→ n ^3 is continuous. (d) Exhibit a function g : N → N
which is not continuous.

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