Question

Prove that if a topological space X has the fixed point property, then X is connected

Answer #1

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A topological space is totally disconnected if the connected
components are all singletons. Prove that any countable metric
space is totally disconnected.

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Give an example of a topological space X that is Hausdorff, but
not regular. Prove that the space X you chose is Hausdorff. Prove
that the space X you chose is not regular.

Prove that if X is a connected Hausdorff space and X has more
than one point, then
X is infinite.
Please show all work and write clear
I am stupid

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

Prove that the product of two connected topological spaces is
connected.

1. a) State the definition of a topological space.
b) Prove that every metric space is a topological space.
c) Is the converse of part (b) true? That is, is every
topological space a metric space? Justify your answer.

Show by counterexample that a connected component of a
topological space is not necessarily open.

Prove that if T is a discrete topological space (i.e., every subset
of T is open) and ∼ is any equivalence relation on T , then the
quotient space T / ∼ is also discrete topological space. [Hint: It
is a very short and straightforward proof.]

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...

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