# Tagged Questions

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

Why $\Omega \neq \bar{\Omega} \setminus \partial \Omega$ ? Can somebody show any counterexample?
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### 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.
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### 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.
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### 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 ...
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### 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...
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### 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 $Y$? ($\sim$ ...
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### 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 ...
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### 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. ...
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### 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 ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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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 $q_n \in ... 0answers 65 views ### Smash products of pointed spaces is really not associative The canonical bijective map$\mathbb{N} \wedge (\mathbb{Q} \wedge \mathbb{Q}) \to (\mathbb{N} \wedge \mathbb{Q}) \wedge \mathbb{Q}$is not an isomorphism of pointed spaces (i.e. homeomorphism), see ... 3answers 100 views ### Give an example of non-normal subspace of a normal space. We know that a closed subspace of normal space is normal. My question was: why should other subspaces not work and then i came up with a counterexample. It is peculiar that any subspace of regular ... 2answers 140 views ### Closed, orientable surface whose genus is very hard to find intuitively I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in$\mathbb{R}^3$... 4answers 939 views ### Example of two open balls such that the one with the smaller radius contains the one with the larger radius. Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ... 1answer 56 views ### Does$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 ...
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 ...