There is not limit to the number of attempts
for each question. However, in order to make the best use of this quiz, you
should give youself sufficient time to think about every question carefully
before submitting an answer. You should not only guess the answer, but
also think about an argument that supports your choice. You will then be
able to compare your justification with the argument in the provided model
solution.
Start Quiz
Which of the following assertions are true?
There is at least one mistake.For example, choice (a) should be
True.
There is at least one mistake.For example, choice (b) should be
False.
There is at least one mistake.For example, choice (c) should be
False.
Consider
with
the indiscrete topology.
There is at least one mistake.For example, choice (d) should be
True.
This is
even true for arbitrary subsets of a Hausdorff space (see Propostion 4.8)
There is at least one mistake.For example, choice (e) should be
True.
This is Proposition 5.9.
Correct!
- True This is Theorem 5.8.
- False Consider
.
- False Consider
with
the indiscrete topology.
- True This is
even true for arbitrary subsets of a Hausdorff space (see Propostion 4.8)
- True This is Proposition 5.9.
Consider subsets
of some Hausdorff space. Which of the following implications do hold?
There is at least one mistake.For example, choice (a) should be
True.
This is one direction of Theorem 5.11.
There is at least one mistake.For example, choice (b) should be
True.
This is the other direction of Theorem 5.11.
There is at least one mistake.For example, choice (c) should be
True.
Take
an open cover
of
. Then
is also an open
cover for
and
, respectively.
Compactness of
and
implies the existence
of finite subcovers
and
of
and
, respectively.
Then
is a finite
subcover for
.
There is at least one mistake.For example, choice (d) should be
False.
Consider
.
There is at least one mistake.For example, choice (e) should be
True.
By
Theorem 5.8 the subsets
and
are closed. Hence, the
intersection
is closed as
well. Then
is also closed in
the subspace topology on
.
As a closed subset of a compact topological space
is
compact as well.
Note, that the Hausdorff property is essential.
There is at least one mistake.For example, choice (f) should be
False.
Consider .
Correct!
- True This is one direction of Theorem 5.11.
- True This is the other direction of Theorem 5.11.
- True Take
an open cover
of . Then
is also an open
cover for and
, respectively.
Compactness of and
implies the existence
of finite subcovers
and of
and
, respectively.
Then is a finite
subcover for .
- False Consider .
- True By
Theorem 5.8 the subsets
and are closed. Hence, the
intersection is closed as
well. Then is also closed in
the subspace topology on .
As a closed subset of a compact topological space
is
compact as well.
Note, that the Hausdorff property is essential.
- False Consider .
Assume
is a continuous map from a compact space to a Hausdorff space. Which of the following
statements are necessarily true? Think carefully about a justification of your answer!
There is at least one mistake.For example, choice (a) should be
True.
This is the definition of continuity of
.
There is at least one mistake.For example, choice (b) should be
True.
This is equivalent to the continuity of
.
There is at least one mistake.For example, choice (c) should be
False.
Take
the
inclusion map. Then
,
which is not open in
.
There is at least one mistake.For example, choice (d) should be
True.
A closed subset
is compact, since
is compact. Then its
image is compact,
since is continuous.
Now, it follows that
is closed, since
is Hausdorff.
Correct!
- True This is the definition of continuity of
.
- True This is equivalent to the continuity of
.
- False Take the
inclusion map. Then ,
which is not open in .
- True A closed subset
is compact, since
is compact. Then its
image is compact,
since is continuous.
Now, it follows that
is closed, since
is Hausdorff.
Every continuous bijection between two bounded and closed subsets of Euclidean
space is a homeomorphism.
Choice (a) is Correct!
Bounded and closed subsets of Euclidean space are
compact by the Heine-Borel Theorem. As subspaces of a Hausdorff space they are
also Hausdorff. Hence, every continuous bijection between those spaces is a
homeomorphism.
Choice (b) is Incorrect.
Bounded and closed subsets of Euclidean space are compact by
the Heine-Borel Theorem. As subspaces of a Hausdorff space they are also Hausdorff.
Hence, every continuous bijection between those spaces is a homeomorphism.
Consider two topologies
and
on the same
topological space
such that
. Which of the following
assertions are true?
There is at least one mistake.For example, choice (a) should be
True.
Indeed, consider
.
Then
.
There is at least one mistake.For example, choice (b) should be
False.
Pick any .
Then .
Correct!
- True Indeed, consider .
Then .
- False Pick any .
Then .
Consider two topologies
and
on the same
topological space
such that
. Which of the following
assertions are true?
There is at least one mistake.For example, choice (a) should be
False.
Take
and
to be the discrete and
to be the indiscrete
topology.
There is at least one mistake.For example, choice (b) should be
True.
Given
two distinct elements
.
If they admit two disjoint open neighbourhoods from
then the same subsets are also disjoint open nieghbourhoods with respect to
, since
.
There is at least one mistake.For example, choice (c) should be
True.
Given
an cover
of
consisting of
elements from
.
Then this is also an open cover with respect to
. Hence, by compactness
of
it admits a
finite subcover.
There is at least one mistake.For example, choice (d) should be
False.
Take
and
to be the discrete
and
to be the
indiscrete topology.
There is at least one mistake.For example, choice (e) should be
False.
Take
and
to be the discrete
and
to be the
indiscrete topology.
There is at least one mistake.For example, choice (f) should be
True.
Note that is a
continuous bijection from a compact space to a Hausdorff space. Hence, it is a homeomorphism.
Therefore .
Correct!
- False Take and
to be the discrete and
to be the indiscrete
topology.
- True Given
two distinct elements .
If they admit two disjoint open neighbourhoods from
then the same subsets are also disjoint open nieghbourhoods with respect to
, since
.
- True Given
an cover of
consisting of
elements from .
Then this is also an open cover with respect to
. Hence, by compactness
of it admits a
finite subcover.
- False Take and
to be the discrete
and to be the
indiscrete topology.
- False Take and
to be the discrete
and to be the
indiscrete topology.
- True Note that is a
continuous bijection from a compact space to a Hausdorff space. Hence, it is a homeomorphism.
Therefore .