# Tagged Questions

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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $B \subset \mathbb{R^3}$, 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.
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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.