In: Advanced Math

(Topology) Prove that the interior of a subset A of Xτ is the union of all τ -open sets contained in A.

(X,τ) is a topological space and A⊆X then a point a∈A is said to be an interior point of A if there exists an open neighbourhood U of a contained in A, that is, there exists a U∈τ with a∈U such that

a∈U⊆A

We call the set of all interior points of A the interior of A and denoted it by int(A) .

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.

Show that any open subset of R (w std. topology) is a countable union of open intervals.What is the objective of this problem and enough to show ?

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.

Let H be the subset of all skew-symmetric matrices in
M3x3
a.) prove that H is a subspace of M3x3 by checking
all three conditions in the definition of subspace.
b.) Find a basis for H. Prove that your basis is actually a
basis for H by showing it is both linearly independent and spans
H.
c.) what is the dim(H)

Prove or disprove if B is a proper subset of A and
there is a bijection from A to B then A is infinite

Topology question:
Prove that a bijection f : X → Y is a homeomorphism if and only
if f and f−1 map closed sets to closed sets.

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.

Let A be an infinite set and let B ⊆ A be a subset. Prove:
(a) Assume A has a denumerable subset, show that A is equivalent
to a proper subset of A.
(b) Show that if A is denumerable and B is infinite then B is
equivalent to A.

Real Analysis: Prove a subset of the Reals is compact if and only
if it is closed and bounded. In other words, the set of reals
satisfies the Heine-Borel property.

Let R be the real line with the Euclidean topology.
(a) Prove that R has a countable base for its topology.
(b) Prove that every open cover of R has a countable
subcover.

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