# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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### Reference for closed map lemma

I would like to have a reference (book, page) for the following version of the closed map lemma: If a continuous function between locally compact Hausdorff spaces is proper (i.e. preimages of compact ...
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### If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded ...
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### Are these subsets homeomorphic?

Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and $[0,1)\times[0,1)$ homeomorphic or not? My attempt: We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ ...
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### Question about the proof that the Hilbert Cube is compact.

Because of the fact that $(1)$ The topological space $[0,1]$ is a continuous image of the Cantor space $(G,T)$. There exists a mapping $\phi_n$ of $(G_n, T_n)$ onto $(I_n, T'_n)$ where, for each ...
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### Introductory Topology Book Recommendation for Economics

Would you please share your 2 cent on book recommendation for introductory topology book to graduate student in Economics. Have exposure to the first half of the yearlong analysis course in the ...
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### Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
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### Embedding and homeomorphism

Suppose there exists an embedding from one topological space into another, and conversely. Is it always true that there is a homeomorphism between the two spaces?
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### Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
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### Question about continuous onto maps of homeomorphic spaces.

If $f:(A,T) \rightarrow (B,T_1)$ is continuous and onto, and $$(A,T) \cong (C,T_2) \land (B,T_1) \cong (D, T_3)$$ $$\Rightarrow \exists g: (C,T_2) \rightarrow (D,T_3)$$ that is continuous and onto.
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### Is the closure of a geodesically convex set convex?

Is the closure of a geodesically convex set convex? If so, is there a simple proof for it? In $\mathbb{R}^n$ there is a simple proof for it through convergent sequences. How should I apply it on ...
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### Question about the composite of a homeomorphism and a continuous onto function.

If $f : (G,T)$ homeomorphically to $(A,T_1)$, and $h: (A,T_1)$ continuously and onto $(C,T_2)$, then is it always the case that, given the composition $g = h \circ f : (G,T) \rightarrow (C,T_3)$, the ...
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### Topology .. cluster points

prove that if x is a cluster point of A unoin B then x is a cluster point of A or B I proved it by contrapositive but I want to prove it with direct proof
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### problem in topology. looking for conditions under which given topology is discrete? [closed]

Let $\tau$ be the topology on $\mathbb{R}$ for which the intervals $[a, b), -\infty < a< b < \infty$, form a base. Let $\sigma$ be a topology on $\mathbb{R}$ such that $\sigma \supseteq \tau$....
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### Existence of sequence of polynomials such that $\lim_{n\to\infty} \int_0^1 |h(x) - p_n(x)|^2 dx = 0$

For a function $h:[0,1] \to \mathbb{R}$: $$h(x) = \begin{cases} 1~~\text{for}~~ x\in[0, \frac12] \\0 ~~\text{for}~~ x\in(\frac12, 1] \end{cases}$$ how could we prove the existence of sequence of ...
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### The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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### Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
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### Proof that a mapping onto $[0,1]$ is continuous.

Let each $(A_i,T_i) = (\{0,2\}, T_{discrete})$ and define $\phi : \prod (A_i, T_i) \rightarrow [0,1]$ with $\phi (<a_1, a_2, ...>) = \sum^{\infty}_{i=1} \frac{a_i}{2^{i+1}}$. In order to show ...
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### When do minimal subcovers always exist - without choice?

In my answer to Doubt in the definition of a compact set, I sketched a proof of the following fact: Suppose $X$ is a topological space such that every open cover of $X$ has a minimal subcover. ...
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### Doubt in the definition of a compact set

It's said a set $A$ is compact if for every finite cover $U$ of $A$ there exists a subset of $U$ which also covers $A$, let's say $U_1$. Assuming $A$ is a compact set, we must be able to find a ...
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### Show finite complement topology is, in fact, a topology

My attempt to prove the following is below: Let X be an infinite set. Show that $\mathscr{T}_1=\{U \subseteq X : U = \emptyset$ or $X\setminus U$ is finite $\}$ My book calls this set the "...
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### Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous, then it is continuous from the right

I'm trying to prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous (where the topology of $\mathbb{R}$ is $\emptyset$, $\mathbb{R}$, and all sets of the form $(-\infty, a)$), then it is continuous ...
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### core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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### Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
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### What does it mean to have a “different topology”?

On a space, I understand the notion of having different metrics on the same space. It is, in layman's terms, different ways of defining a distance but on the same space. But I often see the term "...
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### The topological space $[0,1]$ is a continuous image of the Cantor space question.
Prove that the topological space $[0,1]$ is a continuous image of the Cantor space $(G,T')$. I know that this means to show there exists a function $$(i) f : (G,T') \rightarrow [0,1]$$ such that $f$ ...