Questions tagged [general-topology]
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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.
59,040 questions
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How to glue together edges when constructing the geometric realization $|\mathcal{P}|$ of a polygonal presentation $\mathcal{P}$.
The below excerpt is taken from J. Lee:s "Introduction to Topological Manifolds".
I have a hard time understanding exactly what the "algorithm" for gluing edges together is here. ...
- 1,687
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15
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Showing that the classical groups are topological manifolds [duplicate]
I took a course in differential geometry last semester and I think I understood the basic theory of differential manifolds, but I recently read basic notions about topological manifolds and I’m trying ...
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2
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42
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Show that the inclusion $\phi$ from $l^1$ to $c_0$ is not a topological homomorphism
Let $l^1$ be the space of absolute summable sequences, that is $(x_n)_{n \in \mathbb{N}}$ such that $\forall n \in \mathbb{N}, x_n \in \mathbb{C}$ and $\sum_{n=1}^{\infty} |x_n| \lt \infty$.
Let $c_0$ ...
- 347
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Defining topological properties of functions without homeomorphisms
I started writing this thinking it was going to be more vague, but I ended up kind of half-attempting to answer my own question in the process and now my main question is whether or not my half-...
- 251
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21
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Different $\sigma$-algebras on the space of Borel probability measures on a metric space
This question is intended to distill the core components of this sister question on MathOverflow for increased visibility and transparency.
Let $X$ be a metrizable topological space and $\mathscr B_X$ ...
- 23.7k
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Proving the homotopy equivalence between the real projective plane and the adjunction space made of $D^2$ and $S^2$
The adjunction space is given by $S^2 \cup_f e^2$ where $f:S^1 \to S^1, z \mapsto z^2$ maps antipodal points of $S^1$.
Am I right to picture this as the sphere with the boundary of the disk glued to a ...
- 1
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42
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About open and closed sets of a topological space [closed]
Let $X$ be a topological space.
Let $F_1$ and $F_2$ be two closed and disjoint sets.
Then there exists two open and disjoint sets $A_1$ and $A_2$ such that $F_1 \subseteq A_1$ and $F_2 \subseteq A_2....
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27
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Posets associated to covers
Given a covering $U_i$ of a topological space $X$ for $i$ in some indexing set $S$, there is a poset $P$ whose objects are subsets $I$ of $S$ so that $\cap_{I} U_i$ is non-empty and morphisms are ...
- 1,594
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2
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68
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Iterations of the limit set
I have a doubt on the limit set.
If we call limit set S' the set of all limit points of a given set S, is it by any means true that, regardless of the initial set S, we have S''' = S'' ? In other ...
- 21
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Simplicial maps determined by their vertex maps.
I want to prove the following.
Let $K$ and
$L$ be simplicial complexes. Suppose $f_0:K_0 \to L_0$ is any map with the property
that whenever $\{v_0, \ldots,v_k\}$ are the vertices of a simplex of $K$,...
- 1,687
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1
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42
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Fundamental group calculations on wedges of circles and punctured plane
I want to calculate the fundemental group of the following spaces $X$:
Say $C_n=\{(x,y)\in \mathbb R^2\colon (x-n)^2+y^2=n^2\}$, and $X=\bigcup_{n=1}^{+\infty} C_n$.
Say $C_n=\{(x,y)\in \mathbb R^2\...
- 51
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59
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When, if ever, can plane curves be dense subsets of a unit square?
Definitions of terms used in this question can be found here.
I think (?) that it is clear that an open (i.e. non-closed) (plane) curve can be a dense subset of $[0,1]\times [0,1].$
(Or is this ...
- 22.6k
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Prove that the homomorphism induced by a retraction is an epimorphism
If $r:X\rightarrow A$ is a retraction and $a_0\in A$, then $r_*:\Pi_1(X,a_0)\rightarrow\Pi_1(A,a_0)$ is an epimorphism.
Proof: By definition, $r_*$ is a homomorphism. We now show that $r_*$ is ...
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Given a convex set $A$ in $R^n$, when is Fr $A$ convex?
I am self-studying topology using Viro et al.'s Elementary Topology Problem Textbook. The question above occurs early in the text, and though I have a solution, it feels a little ugly to me, and I ...
- 1,019
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43
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Spaces whose Points violate Homotopy Extension Property
I'm looking for a pair $(X,x)$ consisting of a topological space $X$ with a distinguised point $x \in X$ such that canon inclusion $i_x: x \to X$ is not a cofibration, ie not satisfied homotopy ...
- 8,441
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36
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Prove that the inclusion of the open disk into the closed disk is a compactification and that it is not the Alexandroff compactification. [closed]
Could someone help me demonstrate the following?:
“Prove that the inclusion of the open disk into the closed disk is a compactification and that it is not the Alexandroff compactification.”
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6
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2
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188
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Two types difinition of the distance function
Let $\Omega \subset \mathbb{R}^n$ be a domain, $n \ge 2$. I'm wondering that for any
$x \in \Omega$, do we have
$$
\text{dist}(x,\partial \Omega) = \text{dist}(x, \Omega^c)?
$$
It seems natural. ...
- 359
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27
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Inverse image of closure under perfect open mapping.
Let $f$ be an open and perfect mapping from $X$ to $Y$. Let $A \subset Y$ and let $y \in \overline{A}$, then whether it is true that $f^{-1}(y) \subset \overline{f^{-1}(A)}$.
I have assumed that $y \...
- 441
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2
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99
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Arzela-Ascoli lemma for separable (not necessarily compact) space: Can the subsequence converge to non-continuous function?
In Royden's Real Analysis there is an Arzela-Ascoli Lemma for separable space (which does not need compactness):
$X$ is a separable metric space and $(f_n)$ is an equicontinuous sequence of real ...
- 2,235
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1
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65
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Proposition 10.13, "Attaching a disk".
The excerpt below is taken from J. Lees "Introduction to Topological Manifolds".
In the absolutely last sentence of the second picture, the claim is that $\overline{b} \cdot c \cdot b$ is a ...
- 1,687
4
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1
answer
58
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Homotopy equivalence between $ S^1 $ and a space of quadratic equations with distinct real roots
Let $X = \{(a, b, c) \in \mathbb{R}^3 \mid b^2 > 4ac\}$. I am trying to prove that $X$ has the homotopy type of $S^1$.
This problem arises from an exercise in a topology textbook, where I am ...
- 41
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21
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Selfjoin of simplicial complex [closed]
Prove that $C \ast C \ast C$ is a simply connected topological space for C simplicial complex with r vertices and without simplices of dimension greater than 0.
Is it true that in the conditions above ...
- 21
1
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1
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65
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$[S^n,X]=0\Longrightarrow\pi_n(X,x_0)=0$
Let X be a path connected topological Space, then I want to prove that
$
\forall n\in\mathbb{N}:[S^n,X]=0\Longrightarrow\forall n\in\mathbb{N}:\pi_n(X,x_0)=0
$
Here
$
[S^n,X]:=\{f:S^n\rightarrow X\}/(...
- 43
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46
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Attaching a finite number of cells to finite CW-complex yields a finite CW-complex
As we can see, CW-complex is built from attaching cells in the order of their dimension.
I would like to confirm that, is it true that attaching finite number of cells to a finite CW-complex yields a ...
- 425
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26
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Homotopy Extension Theorem: Is 'Retract of an Open Subset' a Weaker Condition?
${1.4. \textbf{Theorem}}$
Let $Z$ be a closed subspace of a normal space $N$. Let $f: N \to P$ be a continuous map, and let $g: Z \times I \to P$ be a homotopy of $f|Z$. If $g$ extends to a homotopy ...
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Is the real number line $\mathbb{R}$ with the compact complement topology simply connected?
Let $\mathbb{E}^1$ denote $\mathbb{R}$ equipped with the standard Euclidean topology. Let $X$ denote $\mathbb{R}$ with the topology for which a nonempty subset $U \subset \mathbb{R}$ is open if and ...
- 1,893
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68
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Baby Rudin's proof that non-empty Perfect subsets of $\mathbb{R}^k$ are uncountable
I'm translating from the french version but I saw a copy of the proof from the english edition and the french translation seems identical. Anyhow I think I have the terminology right.
Suppose $\...
- 143
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33
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Simplicial complexes and their geometric realization
Might be a silly question but, I’m working on a problem about simplicial complexes and their geometric realization, trying to determine if they are homotopy equivalent to a bouquet of circles. ...
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73
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Why is this not a homotopy equivalence between $S^2$ w/ diameter and $S^1 \vee S^1 \vee S^2 \vee S^2$?
By difference of their fundamental groups, it is obvious that $S^2$ w/ a diameter is not homotopy equivalent to $S^1 \vee S^1 \vee S^2 \vee S^2$. However, I’m not sure what is wrong with my ...
- 1,287
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70
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The closure of $c_{00}$ in $\ell^1$ [closed]
Which is the closure of
$c_{00} = \{(x_n)_n \in \mathbb{R}: \exists N \in \mathbb{N}: \forall n \geq N: x_n = 0\}$ in $\ell^1(\mathbb{R})$?
I know that in $\ell^\infty(\mathbb{F})$ the closure of $C_{...
- 1
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18
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No boundary wave function, quantum cosmology and metrics [closed]
I've been reading up on the No Boundary wave function and how to compute saddle point approximation integrals that do not converge using Picard Lefschetz theory. I see how the calculations work - but ...
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1
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34
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Continuity and Neighborhoods: Validity and Counterexample for the Converse Statement
Let $f$ be a function defined from $\mathbb{R}$ to $\mathbb{R}$. If $f$ is continuous at $x_0$, then there exists a neighborhood $V$ of $x_0$ such that the restriction of $f$ to $V$, denoted $f|_V$, ...
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106
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Why do we need $X\in\mathcal{T}$ for topologies? [duplicate]
A topology is defined as a class $\mathcal{T}$ of subsets of a set $X$ which is closed under arbitrary unions, finite intersection, and such that $X\in\mathcal{T}$. If we study topology, we can note ...
- 457
3
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1
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62
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Homotopy equivalence of open Euclidean sets implies diffeomorphism
Proposition: Let $n\geq 1$ and $~U,V\subseteq\mathbb R^n$ be nonempty, open and (weakly) homotopy equivalent. Then $U$ and $V$ are diffeomorphic.
Intuitively, I think this has to be true. Open subsets ...
- 245
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47
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How was Kuratowski's problem for closure and complementation proven
Kuratowski's problem states that, if $(X,\tau)$ is a topological space and $c,k,i:\mathcal P(X)\to\mathcal P(X)$ are the complementation, closure and interior operator relative to $X$. It is then a ...
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28
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If $B$ is a subset of a connected space $X$, and $C$ is a connected component of $B$, is it true that $\partial C \subseteq \partial B$? [duplicate]
I have the following problem: If $A$ is a subset of a connected space $X$, and $C$ is a connected component of $X \setminus A$, is it true that $\partial C \subseteq \partial A$?
Note that I am not ...
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On the continuity a function given by evaluating compact subsets of continuous functions
Let $B$ be a closed ball in $\mathbb{R}^n$ and write $C(B)$ for the Banach space (with respect to the supremum norm) of the continuous real-valued functions on $B$.
Now given a compact subset $K$ of $...
- 590
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1
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47
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missing step in proof of open subspace of E is E
Let $X$ be un vector subspace of a metric vector space $(E,d)$.
A very classical result is that $X$ open $\Leftrightarrow X=E$.
The proof relies on:
$\vec{0}\in X$
$X$ open $\Rightarrow\exists r\in \...
- 203
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30
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How the K-topology generates a topology for real line?
In Munkres' we define the K-topology as follows:
Let $K$ denote the set of all numbers of the form $1 / n$, for $n \in \mathbb{Z}_{+}$, and let $\mathscr{B}^{\prime \prime}$ be the collection of all ...
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Difference sets of discrete sets and iterations thereof
For $E\subseteq\mathbb{R}$, let us define the “difference set” of $E$ as:
$$
\Delta E := \{x-y : (x,y)\in E^2\}
$$
Furthermore, when $\mathcal{C} \subseteq \mathcal{P}(\mathbb{R})$, let
$$
\Delta[\...
- 5,956
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31
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How to prove the direct product of identification map is not an identification map
How to prove the direct product of identification map is not an identification map? Let ${X}_{1} = \left\lbrack {0, 1}\right\rbrack \times$ $\mathbb{Z}$, on ${X}_{1}$ we have an equivalence ...
- 3,713
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135
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The Hilbert cube cannot be embedded in some Euclidean space $\mathbb R^n$
The Hilbert cube $X=[0,1]^\mathbb N$ with the product topology cannot be embedded in some Euclidean space $\mathbb R^n$ for some finite $n$.
What would be a simple way to show this?
I am particularly ...
- 6,259
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38
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Why is this shape homotopy equivalent to a wedge sum of three circles?
The below image depicts two examples, wherein the arrows denote morphisms that are homotopy equivalences.
I’m trying to understand why, in (I), $X_1$ is homotopy equivalent to $Y_1$. My only intuition ...
- 1,287
3
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35
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Presentation Complex is Homotopy Equivalent to Wedge of Two-Spheres
Let $G$ be a group with presentation $\langle a,b \mid a^3, b^3, (ab)^2\rangle$. Denote the presentation complex of $G$ as $X$. Then the universal cover of $X$, which we denote as $\tilde{X}$, is ...
- 638
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1
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54
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Show that a uniform space is complete iff every Cauchy ultrafilter converges
A uniform space $(X,\Phi)$ Wikipedia
is said to be complete if every Cauchy filter (a filter such that for every $U \in \Phi$ , there exists $A \in \mathcal{F}$ such that $A \times A \subset U$) ...
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3
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1
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61
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Proof $X=\Bbb R/\Bbb Z$ is not homeomorphic to $B$.
I want to show that the space $B$ is not homeomorphic to the space $X$, where
$$
B:=\bigcup_{n \in \mathbb{Z}^{+}} C_n=\left\{(x, y) \in \mathbb{R}^2 \left\lvert\,\left(x-\frac{1}{n}\right)^2+y^2=\...
- 864
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1
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45
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Connection between definition of topological continuity and the quotient topology
Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ is continuous if $f^{-1}(U)$ is open for every open subset $U\subseteq Y$.
If we now require $Y$ to be merely a set and $f$ to be surjective, ...
- 1,621
5
votes
1
answer
36
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Uncountable product of spaces with at least two points is not hereditarily normal
Suppose that $\prod_{i\in I} X_i$ is a product of spaces $X_i$ with $|X|\geq 2$ where $I$ is uncountable.
If all of the spaces $X_i$ are indiscrete, then $\prod_{i\in I} X_i$ is indiscrete, hence ...
- 11.9k
2
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1
answer
94
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homeomorphism between topological manifolds. Invariance of domain.
I am studying topological manifold with boundaries (no smooth structure added to it yet) and I am confused about a definition and what apparently seemed to be a straightforward exercise.
Firstly, my ...
2
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1
answer
40
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Is $\omega_1$ Meta-Lindelöf?
A space is meta-Lindelöf provided every open cover of $X$ has a point-countable open refinement. (A collection $\mathscr V$ of subsets of $X$ is point-countable if each point $x\in X$ belongs to at ...
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