3
votes
2answers
26 views

Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
0
votes
1answer
33 views

Nowhere dense set - coarser vs. finer topology

Let $X$ be a set and let $\tau_1\subseteq\tau_2$ be topologies on $X$. Suppose that $A\subseteq X$ is nowhere dense in $\left(X, \tau_2\right)$. I was wondering if it follows that $A$ is nowhere dense ...
3
votes
1answer
46 views

Closed surjection that does not preserve regularity

Def Map $p\colon X\rightarrow Y$ is perfect if it is a closed surjection and $p^{-1}\left(\left\{y\right\}\right)$ is compact for each $y\in Y$ It is well known that perfect maps preserve regularity, ...
2
votes
1answer
66 views

Does every nonmeasurable set split into a measurable subset and a purely nonmeasurable subset?

Being curious I'm wondering: Suppose you're given a continuous function over a Borel space. Then the preimage of every open is measurable. However, while the preimage of every neighborhood of some ...
0
votes
1answer
32 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
5
votes
2answers
63 views

A Hausdorff space which is not completely regular

My example is, $f : \mathbb{R}^+ \to \mathbb{R}$ defined by: $$f(x) = \begin{cases} x, &\text{if }0 \leq x < 1 \\ \tfrac{1}{x}, &\text{if }x \geq 1. \end{cases}$$ Even though $f(0)=0$ but ...
4
votes
3answers
39 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
0
votes
2answers
33 views

an example of a continuous bijection which is not a homeomorphism [duplicate]

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is ...
0
votes
1answer
34 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
11
votes
1answer
125 views

If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?

Let $X$ be a path connected space and $Y$ be a topological space. Let $f:X\rightarrow Y$ be a function such that for every path $\tau:\mathbb{I}\rightarrow X$ , $f\tau:\mathbb{I}\rightarrow Y$ is ...
0
votes
0answers
20 views

A Lindelof non-scattered space $X$ which is not an extention of $\mathbb R$

Is anyone familier with an example for a Lindelof non-scattered topological space $X$ which is not an extention of $\mathbb R$ (with Euclidean topology). I am looking for an example which is not a ...
-1
votes
1answer
52 views

Why set is not equal its closure minus its boundary? [closed]

Why $ \Omega \neq \bar{\Omega} \setminus \partial \Omega $ ? Can somebody show any counterexample?
1
vote
2answers
55 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
2
votes
1answer
49 views

Can any “relevant” topological spaces be decomposed into an uncountable product?

Can any "relevant", as meaning generally useful topological spaces be decomposed into an uncountable product of other topological spaces with the product topology? Many thanks in advance.
2
votes
0answers
27 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
1
vote
1answer
87 views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
6
votes
0answers
44 views

Is a topological space determined by its components and their quotient?

Given connected topological spaces $X_i$ and a totally disconnected space $Y$, is there a unique topological space $X$ with components homeomorphic to $X_i$ and $X/\sim$ homeomorphic to $Y$? ($\sim$ ...
3
votes
1answer
38 views

Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
3
votes
4answers
90 views

is $\mathbb{R}^2\setminus \{(0,0)\}$ homeomorphic to $S^1$?

I found an exam question asking to prove that a homeomorphism exists but I am quite doubtful that this is true. Can anyone verify this? I can easily prove that the quotient space is homeomorphic. ...
1
vote
1answer
32 views

Quotient of a locally compact space

I am looking for an example of a quotient of a locally compact space that isn't locally compact. Is there a not too complicated example ?
2
votes
2answers
61 views

What are counter examples for these statements?

Question 1. Let $\{T_i\}_{i\in I}$ be a family topologies on a set $X$. Provide an example that $\bigcup T_i$ is not a topology on X. > Question 2. Let $X$ be a compact space ...
4
votes
0answers
26 views

A locally connected metric space of first category

I wonder if there exists a locally connected metric space of first category. I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I ...
7
votes
1answer
104 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
2
votes
0answers
29 views

an example that property $\delta$ does not imply property $\gamma$

In this article, two properties are mentioned at page 153: property $\gamma$: If $\mathcal U$ is an open $\omega$-cover of $X$, then there is a sequence $G_n \in \mathcal U$, with $\underline {Lim} ...
3
votes
1answer
51 views

Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous

I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one. For each of ...
1
vote
1answer
40 views

Compact surjective non injective operator

Let $X$ be an infinite dimensional Banach space. I know that every compact operator $A$ is not bijective or $0\in\sigma(A)$. Fox example the compact operator $A$ defined on $X=C([0,1],\mathbb{R})$ ...
2
votes
1answer
45 views

Is convex set in an ordered set necessarily interval or ray? Munkres 16. 7

The is Problem 7 in Section 16 (page 92) of Munkres' Topology. The problem reads as follows. Let $X$ be an ordered set. If $Y$ is a proper subset of $X$ that is convex in $X$, does it follow ...
3
votes
3answers
105 views

Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space. Can you provide a good example?
-1
votes
1answer
26 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
3
votes
0answers
92 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
1
vote
1answer
64 views

Proofs about continuity and convergence in topological spaces

I'm working on the following exercise: Let $f:(X,T)\to(Y,S)$ and $x\in X$. Prove that if $f$ is continuous at $x$ then if a sequence $\{x_n\}$ converge to $x$ we have $f(\{x_n\})\to f(x)$, show ...
1
vote
2answers
52 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?
3
votes
2answers
63 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
65 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
100 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
140 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 $\mathbb{R}^3$ ...
5
votes
4answers
939 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
56 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
29 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
39 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
25 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
51 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 $X$?
1
vote
1answer
41 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
79 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
1answer
134 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 ...
1
vote
1answer
84 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
62 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
41 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
61 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
44 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 ...