1
vote
2answers
39 views

Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
2
votes
2answers
56 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
3
votes
0answers
26 views

Smash products of pointed spaces is really not associative

The canonical bijective map $\mathbb{N} \wedge (\mathbb{Q} \wedge \mathbb{Q}) \to (\mathbb{N} \wedge \mathbb{Q}) \wedge \mathbb{Q}$ is not an isomorphism of pointed spaces (i.e. homeomorphism), see ...
3
votes
3answers
72 views

Give an example of non-normal subspace of a normal space.

We know that a closed subspace of normal space is normal. My question was: why should other subspaces not work and then i came up with a counterexample. It is peculiar that any subspace of regular ...
6
votes
2answers
109 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
3
votes
4answers
882 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
3
votes
1answer
51 views

Does $X=[0,\omega_1]$ satisfy $S_1(\Omega,\Omega)$?

Definition: An $\omega$-cover of a topological space $X$, is an open cover $\mathcal U$, such that, for any finite set $C \subset X$, there exists an open set $U \in \mathcal U$, such that, $C \subset ...
1
vote
1answer
20 views

The space with countable complement topology (example 20 in “Counterexamples in topology”)

As a continuation to this question: Given the space of countable complement topology on $X$, where $X$ is an uncountable set. (example 20 in "Counterexamples in topology"). We know that $X$ is not ...
0
votes
1answer
37 views

Indiscrete rational extension for $\mathbb R$ (examp 66 in “Counterexamples in topology”)

As a continuation to this question: Let $X$ be the Indiscrete rational extension for $\mathbb R$ (examp 66 page 88 in "Counterexamples in topology"). Let $\langle \mathcal{U}_n: n \in \mathbb{N} ...
0
votes
0answers
18 views

An example of a Lindelöf topological space which is not $\sigma$-compact

I am looking for an example of a Lindelöf topological space which is not $\sigma$-compact. I have looked in Counterexamples in Topology, but, if I am not wrong, all the examples there which meet my ...
2
votes
0answers
47 views

intersection of locally compact Hausdorff topologies.

Are there locally compact Hausdorff topologies $\mathcal T, \mathcal S$ on a set $X$, such that $\mathcal T\cap \mathcal S$ is a Hausdorff but not locally compact topology on $X$?
0
votes
0answers
31 views

Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
4
votes
1answer
67 views

Hard to find counterexample for $\partial (\partial A) = \partial A$

In an exercise I've proven that $\partial(\partial A) \subset \partial A$, for any $A\subset X$, where $X$ is a topological space and $\partial$ in this case stands for the boundary. Apparently, in ...
5
votes
0answers
98 views

Is there a nonempty open bounded subset of plane whose boundary contains no 1 dimensional interval?

Someone asked a question here which hasn't received a correct answer because everyone seems to be misinterpreting the question. I would like to ask the question again. Does there exist a nonempty ...
0
votes
1answer
55 views

Does sequential compactness imply countable compactness?

Let $X$ be a topological space which is sequentially compact. Does this imply that $X$ is countably compact? Thank you!
1
vote
1answer
56 views

Let $A\subset X$; let $f:A\to Y$ be continuous; let $Y$ be Hausdorff. Is there an example where there is no continuous function for $g$?

Let $A\subset X$; let $f:A\to Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g:\mathrm{cl}(A)\to Y$, then $g$ is uniquely determined by $f$. Is ...
2
votes
1answer
35 views

Cartesian Product of Linearly ordered space and an example

The base of the cartesian product of linearly ordered spaces $A$ and $B$ is the form $\{U\times V : U\text{ open in }A, B\text{ open in }B\}$. By using this, we condiser the Long Line Topology which ...
2
votes
1answer
50 views

Is there a regular not completely regular space which is not a corkscrew?

While studying some examples of regular spaces which are not completely regular, I came across Steen's and Seebach's "Counterexamples in Topology". In this book, after searching for examples, I only ...
3
votes
1answer
40 views

Properties of the indiscrete rational extension of $\mathbb{R}$

Let $X = \mathbb R$ equipped with the topology generated by open intervals of the form $(a,b)$ and set of the form $(a,b)\cap \mathbb Q.$ Then $X$ is regular. $X$ is normal $X$ \ $\mathbb Q$ is ...
6
votes
1answer
44 views

Open covers of a topological space $X$

An open cover $\mathcal U$ of a topological space $X$, is called An $\omega$-cover, if every finite subset of $X$, is contained in a member of $\mathcal U$. A $\gamma$-cover if it is infinite and ...
1
vote
1answer
43 views

Upper half plane with irrational slope topology.

Let $X$ be the upper half plane $\{(p,q);p,q \in \mathbb Q, q\geq 0 \}$ with the Irrational slope topology. Can we say that the set $$\{(p,q);p,q \in \mathbb Q, q> 0 \}$$ is a closed set of $X$. I ...
0
votes
0answers
29 views

The space Thomas's Planck from “counterexamples in topology”

Anyone knows where can I get a description of the space " Thomas's Planck "? It is mentioned as example 93 in the book "counterexamples in topology" but there is no description of that space there. ...
2
votes
1answer
71 views

Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
3
votes
0answers
44 views

Lindelöf property and $\omega$-covers

Let $X$ be a Lindelöf topological space. Does this imply that every $\omega$-cover has a countable subcover which is also an $\omega$-cover? if not, is there an example of a topological Lindelöf space ...
0
votes
1answer
17 views

Are there an completely regular, non-lindeloff spaces with only constant real valued functions?

Is there an example for a topological space, which is not lindeloff, but is completely regular, on which continuous real valued functions, are, constant for all $x \in X$, or, constant for all, exapt ...
7
votes
1answer
121 views

Is Thomas' Corkscrew completely regular? (from Counterexamples in Topology)

In Example 94 of Counterexamples in Topology. In the example itself, it is written that the space is completely regular. But in the appendix at the end of the book, it is written that the space is ...
1
vote
1answer
58 views

prove or disprove that $CL(A∩B)=CL(A)∩CL(B)$

Suppose $X,τ$ a topological space. If $A$ and $B$ are any two subsets of $X$ prove or disprove that $CL(A∩B)=CL(A)∩CL(B)$ I know that closed sets is closed under intersection, however I still got the ...
1
vote
1answer
18 views

How can a bounded subspace of the left order topology be compact?

I want to show that every bounded set equipped with the left order topology is compact. This is a statement I found on a wikipedia page and appearently it is lifted from the book Counterexamples in ...
5
votes
2answers
49 views

Two non-homeomorphic connected, hausdorff, locally compact spaces whose one-point compactifications are homeomorphic

I'm looking for two non-homeomorphic connected, Hausdorff, locally compact spaces whose one-point compactifications are homeomorphic. Without the connectedness property this is easy, for example: ...
2
votes
1answer
63 views

Intuitively confusing example of open set and topology on Real Line

My question occurs when I see the problem that show that every continuous function is borel measurable. I know that for topological space $(X,\tau)$, we define that "open sets" means sets in topology. ...
2
votes
3answers
68 views

An example of a non first countable Fréchet-Urysohn space?

As the head title says, I need a Fréchet-Urysohn space but not first countable, (on the way, a good Text book to follow). Thanks.
4
votes
1answer
48 views

Topological space in which there are no close and compacts subsets (except for the empty set)

Any example of those topological spaces? I cant think of no one :S I think it must be infinite and it must not be T2, but no idea how to find one.
1
vote
0answers
40 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
2
votes
2answers
52 views

K topology: Examples

Why would the interval $(-3,1)$ be open in the $k$-topology? (I'm using Munkres). Can I have some other examples of intervals in $k$-topology? What exactly does $(a,b)$ $\cup$ $(a,b)-k$ for ...
9
votes
1answer
105 views

Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?

The particular point topology on any set is connected, but on removing the particular point, the complement is discrete, and hence totally disconnected. Although this is not even $T^1$, Cantor's leaky ...
4
votes
0answers
108 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
2
votes
0answers
25 views

A question on the classical Mrowka space

Definition: A space $X$ is $\Delta$-normal if for every $A \subset X^2 \setminus \Delta_X$ closed in $X^2$ there exist disjoint open $U$ and $V$ in $X^2$ such that $A \subset U$ and $\Delta_X \subset ...
0
votes
1answer
46 views

The Tangent Disc Topology is developable

A well-known example of a Moore space is the Tangent Disc Topology. I want to show that the Tangent Disc Topology is a developable space, i.e. it has a development. But I could not find the proof of ...
2
votes
2answers
50 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
1
vote
1answer
48 views

Closed sets, boundary, topology.

Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not. I ...
4
votes
1answer
80 views

Does “locally connected and path-connected” imply locally path-connected?

Some friends and I discovered this question when we were studying for an exam and were trying to find examples for all combinations of topological properties we had seen in the course so far. One we ...
0
votes
2answers
32 views

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$

$B \subset \mathbb{R^3}$, such that $B = \rm{Fr} (\Omega)$ Where $\Omega = \{(x,y,z) : z \leq 1\}$ and $\rm{Fr}(\Omega)$ means the $\Omega$'s boundary.
1
vote
2answers
29 views

Example of a continuous bijective function on and to the closure of the complex numbers, with an inverse that is not continuous?

Note that by the closure of the complex number, I mean the union of the complex numbers and infinity. I have been stumbling over this questions for a wile now, and I understand many examples of this ...
13
votes
1answer
169 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
2
votes
1answer
98 views

A question on spaces with a point countable base?

Is there a normal Hausdorff space with a point countable base and a dense Lindelöf subspace which is not second countable? Thanks for your help.
0
votes
2answers
194 views

Open and Closed Quotient Maps.

a) Find a subset $A$ of $\Bbb{R}$ such that the quotient map $p: \Bbb{R} \rightarrow \Bbb{R}/A$ is not open. If we let $A= \Bbb{Q}$, then we can see that $(0,1)$ is open in $\Bbb{R}$. But if we ...
1
vote
0answers
32 views

Another question on spaces with calibre-$\aleph_1$

Let $X$ be a strongly monotonically monolithic space with calibre-$\aleph_1$. Must $X$ be Lindelof? I know $e(X)=l(X)$ for a strongly monotonically monolithic space. So to prove that $X$ is ...
4
votes
1answer
53 views

A question on spaces with calibre-$\aleph_1$

Suppose that $X$ is the $T_1$ space with $k$-in-countable base and $\aleph_1$ is a caliber of $X$. Must $X$ be second countable? Thanks for any help. A topological space has calibre $\aleph_1$ if ...
2
votes
1answer
51 views

Another question on second countable spaces

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand. Thanks for your help.
4
votes
1answer
67 views

Examples of spaces in which the closure of any path connected set is path connected.

Are there (non-trivial) examples of topological spaces in which the closure of any path connected set is path connected? If so, are there any far reaching topological consequences of this property? ...