Questions tagged [lindelof-spaces]

A Lindelöf space is a topological space in which every open cover has a countable subcover.

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Is a locally compact hereditarily Lindelof Hausdorff space first countable?

Is a locally compact hereditarily Lindelof Hausdorff space first countable? I was recently told that it is but I can't find any reference to what I would have thought would be a standard fact if it is ...
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21 views

Paracompact Hausdorff Space with Dense Lindelof subset is Lindelof

Let $X$ be a Paracompact Hausdroff space with a dense subset $A$ which is Lindelöf. Then, $X$ is Lindelof I've written down my attenpt below - As per the hint in the problem, as a paracompact $T_2$ ...
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1answer
20 views

Embedding Regular Lindelöf Space in a Hausdorff Space

Preliminary definition - $G_{\delta}$ Closed: A set $A$ is $G_{\delta}$ closed if each point $x\not\in A$ is contained in a $G_{\delta}$ set disjoint from $A$. Willard states that - A regular space ...
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Prob. 14, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: If $X$ is Lindelof and $Y$ is compact, then $X\times Y$ is Lindelof

Here is Prob. 14, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is Lindelof and $Y$ is compact, then then $X \times Y$ is Lindelof. My Attempt: Let $X$ and $Y$...
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Prob. 11, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: A continuous image of a Lindelof (separable) space is Lindelof (separable)

Here is Prob. 11, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Let $f \colon X \rightarrow Y$ be continuous. Show that if $X$ is Lindelof, or if $X$ has a countable dense subset, ...
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Prob. 5 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizable Lindelof space has a countable basis

Here is Prob. 5, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: (a) Show that every metrizable space with a countable dense subset has a countable basis. (b) Show that every ...
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$Y\subset X$ is Lindelof iff Every $X$-open cover of $Y$ has a countable $X$-open subcover. [duplicate]

Suppose $(X,\tau)$ is a topological space and $A\subset X$,suppose $\{G_\alpha\}$ is an open cover of $A$ i.e. $A\subset \cup G_\alpha$,does there exist a countable subcover $\{G_{\alpha_n}\}$ of $A$....
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Second Countability $\implies$ Lindelöf property(Proof verification)

I am a beginner in metric space course.Recently I have learnt the terms second countability and Lindelöfness of a space(metric/topological).Now I have proved that second countability implies ...
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2answers
78 views

Classifying Hausdorff spaces

Can a topological space be Hausdorff and separable, but neither Lindelof nor first countable? Can a topological space be Hausdorff and Lindelof, but neither separable nor first countable? Can a ...
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Almost Lindelöf spaces.

I'm working on the study of almost Lindelöf spaces and I'm stuck searching a counterexample. First, the definition. Let $X$ be a topological space. We say that $X$ is an almost Lindelöf space if ...
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Munkres 37.3 maximal set with respect to the countable intersection property.

The author says, "If $X$ is a set and $\mathbb A$ be the superset consisting of all collections $\mathscr B$ of subsets of $X$ such that $\mathscr A \subset \mathscr B$ and $\mathscr B$ has the ...
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1answer
28 views

Countable $\varepsilon$-net of measurable sets

Let $I=[0,1]$, $\mathcal{A}$ is the family that contains all the Lebesgue measurable sets of $I$, for any $A_1,A_2\in \mathcal{A}$, we define a metric $$d(A_1,A_2)=\int |1_{A_1}-1_{A_2}|.$$ A $\...
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242 views

Lindelöf and separable metric space [duplicate]

Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?
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1answer
81 views

Proof for $\mathbb{R}_{l}$ is Lindelof

In Munkers there is a proof that $\mathbb{R}_{l}$ is Lindelof. $\mathcal{A}$ be the collection of basis elements of the form $\{ [a_{\alpha},b_{\alpha}) : \alpha \in J \}$ be a covering of $\mathbb{R}$...
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Show that a Regular space $X$ is Lindelöf if it has a locally finite open cover $\mathcal{U}$ such that $\rm{Fr}\,U$ has the Lindelöf property

Show that if a regular space $X$ has a locally finite open cover $\mathcal{U}$ such that $\rm{Fr}\,U$ has the Lindelöf property for all $U\in\mathcal{U}$, then the space X is also a Lindelöf Space. ...
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1answer
119 views

Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.

Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable. My attempt: Let $d$ be the metric that induces the topology $\mathcal{T}$. For every $n \geq ...
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2answers
171 views

Show that if a topological space has at most countable basis, then the space is separable and Lindelöf

Let $(X,\mathcal{T})$ be a topological space. Show that if $(X, \mathcal{T})$ has countable base, it is separable (a) and Lindelöf (b) My attempt: Let $\mathcal{B}$ be a countable basis of the ...
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695 views

Proof by Contradiction--Product of Lindelöf Spaces is Lindelöf

Consider the three statements: (i) If $X$ is a set and $\mathcal{A}$ is a collection of subsets of $X$ having the countable intersection property, then there is collection $\mathcal{D}$ of subsets of ...
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1answer
391 views

Countable subspace of a Lindelöf space

Let $X$ be a Lindelöf space, and $E \subset X$. I am interested in the following property. (1) $E$ is countable $\Leftrightarrow$ $\forall x \in X, \exists U_x$ open set such that $x \in U_x$ ...
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3answers
617 views

Every regular Lindelöf space is second countable.

We know that any second countable space is a Lindelöf space but not conversely. Is the following can be taken as a modified converse? Every regular Lindelöf space is second countable. I have ...
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Closed subsets of Lindelöf spaces are Lindelöf

Let $X$ be Lindelöf and let $A \subseteq X$ be closed. Show that it follows that $A$ is Lindelöf. That is, we want to show for every open cover of $A$, there is a countable subcover. Note: (A space ...