On 𝜋-Images of Locally Separable Metric Spaces

Abstract

We characterize 𝜋-images of locally separable metric spaces by means of covers having 𝜋-property. As its application, we obtain characterizations of compact-covering (sequence-covering, pseudo-sequence-covering, and sequentially quotient) 𝜋-images of locally sparable metric spaces.

1. Introduction

To determine what spaces are the images of “nice” spaces under “nice” mappings is one of the central questions of general topology in [1]. In the past, many noteworthy results on images of metric spaces have been obtained. For a survey in this field, see [2], for example. A characterization for a quotient compact image of a locally separable metric space is obtained in [3]. Also, such a quotient image is precisely a pseudo-sequence-covering quotient compact image of a locally separable metric space [4]. Recently, 𝜋-images of metric spaces cause attention once again in [5, 6]. It is known that a space is a compact-covering (resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient) 𝜋-image of a metric space if and only if it has a point-star network consisting of 𝑐𝑓𝑝-covers (resp., 𝑐𝑠-covers, 𝑤𝑐𝑠-covers, 𝑐𝑠^∗-covers) [5–7]. In a personal communication, the first author of [6] informs that it seems to be difficult to obtain “nice” characterizations of “nice” images of locally separable metric spaces (instead of metric or locally compact metric domains). Thus, we are interested in the following question.

Question 1.1. How are compact-covering (resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient) 𝜋-images of locally sparable metric spaces characterized?

In this paper, we characterize compact-covering (resp., sequence-covering, pseudo-sequence-covering, sequentially-quotient) 𝜋-images of locally separable metric spaces by means of 𝑐𝑓𝑝-covers (resp., 𝑐𝑠-covers, 𝑤𝑐𝑠-covers, 𝑐𝑠^∗-covers) for compact subsets (resp., convergent sequences) in a space and covers having 𝜋-property to answer Question 1.1 completely. As applications of these results, we get characterizations on quotient 𝜋-images of locally separable metric spaces.

Throughout this paper, all spaces are assumed to be regular and 𝑇₁, all mappings are assumed continuous and onto, a convergent sequence includes its limit point, ℕ denotes the set of all natural numbers. Let 𝑓∶𝑋→𝑌 be a mapping, 𝑥∈𝑋, and 𝒫 be a collection of subsets of 𝑋, we denote st(𝑥,𝒫)=⋃{𝑃∈𝒫∶𝑥∈𝑃}, 𝒫_𝑥={𝑃∈𝒫∶𝑥∈𝑃}, ⋃𝒫=⋃{𝑃∶𝑃∈𝒫}, and 𝑓(𝒫)={𝑓(𝑃)∶𝑃∈𝒫}. We say that a convergent sequence {𝑥_𝑛∶𝑛∈ℕ}∪{𝑥} converging to 𝑥 is eventually (resp., frequently) in 𝐴 if {𝑥_𝑛∶𝑛≥𝑛₀}∪{𝑥}⊂𝐴 for some 𝑛₀∈ℕ (resp., {𝑥_𝑛𝑘∶𝑘∈ℕ}∪{𝑥}⊂𝐴 for some subsequence {𝑥_𝑛𝑘∶𝑘∈ℕ} of {𝑥_𝑛∶𝑛∈ℕ}). For terms which are not defined here, please refer to [2, 8].

Let 𝒫 be a collection of subsets of a space 𝑋, and 𝐾 be a subset of 𝑋.

𝒫 is a cover for 𝐾 in 𝑋, if 𝐾⊂⋃𝒫. When 𝐾=𝑋, a cover for 𝐾 in 𝑋 is a cover of 𝑋 [8]. A cover 𝒫 for 𝑋 is a compact cover if all members of 𝒫 are compact.

For each 𝑥∈𝑋, 𝒫 is a network at 𝑥 if 𝑥∈𝑃 for every 𝑃∈𝒫, and if 𝑥∈𝑈 with 𝑈 open in 𝑋, there exists 𝑃∈𝒫 such that 𝑥∈𝑃⊂𝑈.

𝒫 is a 𝑘-𝑐𝑜𝑣𝑒𝑟 for 𝐾 in 𝑋, if for each compact subset 𝐻 of 𝐾 there exists a finite subfamily ℱ of 𝒫 such that 𝐻⊂⋃ℱ. When 𝐾=𝑋, a 𝑘-cover for 𝐾 in 𝑋 is a 𝑘-𝑐𝑜𝑣𝑒𝑟 for 𝑋.

𝒫 is a 𝑐𝑓𝑝-𝑐𝑜𝑣𝑒𝑟 for 𝐾 in 𝑋, if for each compact subset 𝐻 of 𝐾 there exists a finite subfamily ℱ of 𝒫 such that 𝐻⊂⋃{𝐶_𝐹∶𝐹∈ℱ}, where 𝐶_𝐹 is closed and 𝐶_𝐹⊂𝐹 for every 𝐹∈ℱ. Note that such an ℱ is a full cover in the sense of [9]. When 𝐾=𝑋, a 𝑐𝑓𝑝-cover for 𝐾 in 𝑋 is a 𝑐𝑓𝑝-𝑐𝑜𝑣𝑒𝑟 for 𝑋 [10].

𝒫 is a 𝑐𝑠-𝑐𝑜𝑣𝑒𝑟 for 𝐾 in 𝑋 (resp., 𝑐𝑠^∗-𝑐𝑜𝑣𝑒𝑟 for 𝐾 in 𝑋), if for each convergent sequence 𝑆 in 𝐾, 𝑆 is eventually (resp., frequently) in some 𝑃∈𝒫. When 𝐾=𝑋, a 𝑐𝑠-cover for 𝐾 in 𝑋 (resp., 𝑐𝑠^∗-cover for 𝐾 in 𝑋) is a 𝑐𝑠-𝑐𝑜𝑣𝑒𝑟 for 𝑋 [11] (resp., 𝑐𝑠^∗-𝑐𝑜𝑣𝑒𝑟 for 𝑋 [12]), or a condition (𝑐₃) (resp., (𝑐₂)) [5].

𝒫 is a 𝑤𝑐𝑠-𝑐𝑜𝑣𝑒𝑟 for 𝐾 in 𝑋 if for each convergent sequence 𝑆 converging to 𝑥 in 𝐾 there exists a finite subfamily ℱ of 𝒫_𝑥 such that 𝑆 is eventually in ⋃ℱ. When 𝐾=𝑋, a 𝑤𝑐𝑠-cover for 𝐾 in 𝑋 is a 𝑤𝑐𝑠-𝑐𝑜𝑣𝑒𝑟 [7].

It is clear that if 𝒫 is a cover (resp., 𝑘-cover, 𝑐𝑓𝑝-cover, 𝑐𝑠-cover, 𝑤𝑐𝑠-cover, 𝑐𝑠^∗-cover), then 𝒫 is a cover (resp., 𝑘-cover, 𝑐𝑓𝑝-cover, 𝑐𝑠-cover, 𝑤𝑐𝑠-cover, 𝑐𝑠^∗-cover) for 𝐾 in 𝑋. Remark 1.2. (1) Closed 𝑘-cover for 𝐾 in 𝑋⇒𝑐𝑓𝑝-cover for 𝐾 in 𝑋⇒𝑘-cover for 𝐾 in 𝑋;
(2) 𝑐𝑓𝑝-cover for 𝐾 in 𝑋, or 𝑐𝑠-cover for 𝐾 in 𝑋⇒𝑤𝑐𝑠-cover for 𝐾 in 𝑋⇒𝑐𝑠^∗-cover for 𝐾 in 𝑋.

For each 𝑛∈ℕ, let 𝒫_𝑛 be a cover for 𝑋. {𝒫_𝑛∶𝑛∈ℕ} is a refinement sequence for 𝑋 if 𝒫_𝑛+1 is a refinement of 𝒫_𝑛 for each 𝑛∈ℕ. A refinement sequence for 𝑋 is a refinement of 𝑋 in the sense of [4].

Let {𝒫_𝑛∶𝑛∈ℕ} be a refinement sequence for 𝑋. {𝒫_𝑛∶𝑛∈ℕ} is a point-star network for 𝑋, if {st(𝑥,𝒫_𝑛)∶𝑛∈ℕ} is a network at 𝑥 for each 𝑥∈𝑋. Note that this notion is used without the assumption of a refinement sequence in [13], and in [5] ⋃{𝒫_𝑛∶𝑛∈ℕ} is a 𝜎-𝑠𝑡𝑟𝑜𝑛𝑔 network for 𝑋.

A cover {𝑋_𝜆∶𝜆∈Λ} for 𝑋 is called to have 𝜋-𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 if each 𝑋_𝜆 has a refinement sequence {𝒫_𝜆,𝑛∶𝑛∈ℕ} of countable covers for 𝑋_𝜆, and for each 𝑥∈𝑈 with 𝑈 open in 𝑋, there is 𝑛∈ℕ such that{st(𝑥,𝒫_𝜆,𝑛)∶𝜆∈Λwith𝑥∈𝑋_𝜆}⊂𝑈. (1.1)

Let {𝒫_𝑛∶𝑛∈ℕ} be a point-star network for a space 𝑋. For every 𝑛∈ℕ, put 𝒫_𝑛={𝑃_𝛼∶𝛼∈𝐴_𝑛}, and 𝐴_𝑛 is endowed with discrete topology. Put𝑀={𝑎=(𝛼_𝑛)∈_𝑛∈ℕ𝐴_𝑛∶{𝑃_𝛼𝑛∶𝑛∈ℕ}formsanetworkatsomepoint𝑥_𝑎in𝑋}. (1.2)Then 𝑀, which is a subspace of the product space ∏_𝑛∈ℕ𝐴_𝑛, is a metric space with metric 𝑑 described as follows.

Let 𝑎=(𝛼_𝑛),𝑏=(𝛽_𝑛)∈𝑀. If 𝑎=𝑏, then 𝑑(𝑎,𝑏)=0. If 𝑎≠𝑏, then 𝑑(𝑎,𝑏)=1/(min{𝑛∈ℕ∶𝛼_𝑛≠𝛽_𝑛}).

Define 𝑓∶𝑀→𝑋 by choosing 𝑓(𝑎)=𝑥_𝑎, then 𝑓 is a mapping, and (𝑓,𝑀,𝑋,{𝒫_𝑛}) is a Ponomarev's system [14], and if without the assumption of a refinement sequence in the notion of point-star networks, then (𝑓,𝑀,𝑋,{𝒫_𝑛}) is a Ponomarev's system in the sense of [13].

Let 𝑓∶𝑋→𝑌 be a mapping.

𝑓 is a compact-covering mapping [15], if every compact subset of 𝑌 is the image of some compact subset of 𝑋.

𝑓 is a sequence-covering mapping [16], if every convergent sequence of 𝑌 is the image of some convergent sequence of 𝑋.

𝑓 is a pseudo-sequence-covering mapping [5], if every convergent sequence of 𝑌 is the image of some compact subset of 𝑋.

𝑓 is a subsequence-covering mapping [3], if for every convergent sequence 𝑆 of 𝑌, there is a compact subset 𝐾 of 𝑋 such that 𝑓(𝐾) is a subsequence of 𝑆.

𝑓 is a sequentially-quotient mapping [17], if for every convergent sequence 𝑆 of 𝑌, there is a convergent sequence 𝐿 of 𝑋 such that 𝑓(𝐿) is a subsequence of 𝑆.

𝑓 is a pseudo-open mapping [1], if 𝑦∈int𝑓(𝑈) whenever 𝑓⁻¹(𝑦)⊂𝑈 with 𝑈 open in 𝑋.

𝑓 is a 𝜋-𝑚𝑎𝑝𝑝𝑖𝑛𝑔 [1], if for every 𝑦∈𝑌 and for every neighborhood 𝑈 of 𝑦 in 𝑌, 𝑑(𝑓⁻¹(𝑦),𝑋−𝑓⁻¹(𝑈))>0, where 𝑋 is a metric space with a metric 𝑑.

Let 𝑋 be a space. We recall that 𝑋 is sequential [18], if a subset 𝐴 of 𝑋 is closed if and only if any convergent sequence in 𝐴 has a limit point in 𝐴. Also, 𝑋 is Fréchet (or Fréchet Urysohn) if for each 𝑥∈𝐴, there exists a sequence in 𝐴 converging to 𝑥.

Note that, for a mapping 𝑓∶𝑋→𝑌, 𝑓 is compact-covering or sequence-covering ⇒𝑓 is pseudo-sequence-covering ⇒𝑓 is subsequence-covering [4]. Also, 𝑓 is quotient if and only if 𝑓 is subsequence-covering for 𝑌 being sequential [3].

2. Main Results

Lemma 2.1. Let 𝒫 be a countable cover for a convergent sequence 𝑆 in a space 𝑋. Then the following are equivalent:
(1)𝒫 is a 𝑐𝑓𝑝-cover for 𝑆 in 𝑋;(2)𝒫 is a 𝑤𝑐𝑠-cover for 𝑆 in 𝑋;(3)𝒫 is a 𝑐𝑠^∗-cover for 𝑆 in 𝑋.

Proof. (1) ⇒ (2) ⇒ (3). By Remark 1.2.
(3) ⇒ (1). Let 𝐻 be a compact subset of 𝑆. We can assume that 𝐻 is a subsequence of 𝑆. Since 𝒫 is countable, put 𝒫_𝑥={𝑃_𝑛∶𝑛∈ℕ}, where 𝑥 is the limit point of 𝑆. Then 𝐻 is eventually in ⋃_𝑛≤𝑘𝑃_𝑛 for some 𝑘∈ℕ. If not, then for any 𝑘∈ℕ, 𝐻 is not eventually in ⋃_𝑛≤𝑘𝑃_𝑛. So, for every 𝑘∈ℕ, there exists 𝑥_𝑛𝑘∈𝑆−⋃_𝑛≤𝑘𝑃_𝑛. We may assume 𝑛₁<𝑛₂<⋯<𝑛_𝑘−1<𝑛_𝑘<𝑛_𝑘+1<⋯. Put 𝐻^′={𝑥_𝑛𝑘∶𝑘∈ℕ}∪{𝑥}, then 𝐻^′ is a subsequence of 𝑆. Since 𝒫 is a 𝑐𝑠^∗-cover for 𝑆 in 𝑋, there exists 𝑚∈ℕ such that 𝐻^′ is frequently in 𝑃_𝑚. This contradicts to the construction of 𝐻^′. So 𝐻 is eventually in ⋃_𝑛≤𝑘𝑃_𝑛 for some 𝑘∈ℕ. It implies that 𝒫 is a 𝑐𝑓𝑝-cover for 𝑆 in 𝑋.

The following lemma is routinely shown, so we omit the proof.

Lemma 2.2. Let 𝑓∶𝑋→𝑌 be a mapping.
(1) If 𝒫 is a 𝑘-cover for a compact set 𝐾 in 𝑋, then 𝑓(𝒫) is a 𝑘-cover for 𝑓(𝐾) in 𝑌.(2)If 𝒫 is a 𝑐𝑠-cover for a convergent sequence 𝑆 in 𝑋, then 𝑓(𝒫) is a 𝑐𝑠-cover for 𝑓(𝑆) in 𝑌.

Same as [6, Lemma 2.2(2)(ii)], we get the following.

Lemma 2.3. Let (𝑓,𝑀,𝑋,{𝒫_𝑛}) be a Ponomarev's system. For a convergent sequence 𝑆 of 𝑋, if 𝒫_𝑛 is a 𝑐𝑠-cover for 𝑆 in 𝑋 for each 𝑛∈ℕ, then there exists a convergent sequence 𝐿 of 𝑀 such that 𝑓(𝐿)=𝑆.

Proof. Put 𝑆={𝑥_𝑖∶𝑖∈ℕ}∪{𝑥}, where 𝑥 is the limit point. For each 𝑛∈ℕ, since 𝒫_𝑛 is a 𝑐𝑠-cover for 𝑆, 𝑆 is eventually in some 𝑃_𝛼𝑛∈𝒫_𝑛. For each 𝑖∈ℕ, if 𝑥_𝑖∈𝑃_𝛼𝑛, let 𝛼_𝑛,𝑖=𝛼_𝑛; if 𝑥_𝑖∉𝑃_𝛼𝑛, pick 𝛼_𝑛,𝑖∈𝐴_𝑛 such that 𝑥_𝑖∈𝑃_{𝛼𝑛,𝑖}. Thus there exists 𝑖_𝑛∈ℕ such that 𝛼_𝑛,𝑖=𝛼_𝑛 for all 𝑖>𝑖_𝑛. So {𝛼_𝑛,𝑖∶𝑖∈ℕ} converges to 𝛼_𝑛. For each 𝑖∈ℕ, put 𝑎_𝑖=(𝛼_𝑛,𝑖)∈∏_𝑛∈ℕ𝐴_𝑛, and put 𝑎=(𝛼_𝑛)∈∏_𝑛∈ℕ𝐴_𝑛. Then 𝑎_𝑖∈𝑓⁻¹(𝑥_𝑖), 𝑎∈𝑓⁻¹(𝑥), and 𝐿={𝑎_𝑖∶𝑖∈ℕ}∪{𝑎} converges to 𝑎. It implies that 𝐿 is a convergent sequence of 𝑀 and 𝑓(𝐿)=𝑆.

Proposition 2.4. The following are equivalent for a space 𝑋.
(1)𝑋 is a 𝜋-image of a locally separable metric space,(2)𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property.

Proof. (1) ⇒ (2). Let 𝑓∶𝑀→𝑋 be a 𝜋-mapping from a locally separable metric space 𝑀 with metric 𝑑 onto 𝑋. Since 𝑀 is a locally separable metric space, 𝑀=⊕_𝜆∈Λ𝑀_𝜆, where each 𝑀_𝜆 is a separable metric space by [8, 4.4.F]. For each 𝜆∈Λ, let 𝐷_𝜆 be a countable dense subset of 𝑀_𝜆, and put 𝑓_𝜆=𝑓|_𝑀𝜆 and 𝑋_𝜆=𝑓_𝜆(𝑀_𝜆). For each 𝑎∈𝑀_𝜆 and 𝑛∈ℕ, put 𝐵(𝑎,1/𝑛)={𝑏∈𝑀_𝜆∶𝑑(𝑎,𝑏)<1/𝑛}, ℬ_𝜆,𝑛={𝐵(𝑎,1/𝑛)∶𝑎∈𝐷_𝜆}, and 𝒫_𝜆,𝑛=𝑓_𝜆(ℬ_𝜆,𝑛). It is clear that {𝒫_𝜆,𝑛∶𝑛∈ℕ} is a sequence of countable covers for 𝑋_𝜆, and 𝒫_𝜆,𝑛+1 is a refinement of 𝒫_𝜆,𝑛 for every 𝑛∈ℕ. We will prove that 𝜋-property is satisfied.
For each 𝑥∈𝑈 with 𝑈 open in 𝑋. Since 𝑓 is a 𝜋-mapping, 𝑑(𝑓⁻¹(𝑥),𝑀−𝑓⁻¹(𝑈))>2/𝑛 for some 𝑛∈ℕ. Then, for each 𝜆∈Λ with 𝑥∈𝑋_𝜆, we get 𝑑(𝑓_𝜆⁻¹(𝑥),𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆))>2/𝑛, where 𝑈_𝜆=𝑈∩𝑋_𝜆. Let 𝑎∈𝐷_𝜆 and 𝑥∈𝑓_𝜆(𝐵(𝑎,1/𝑛))∈𝒫_𝜆,𝑛. We will prove that 𝐵(𝑎,1/𝑛)⊂𝑓_𝜆⁻¹(𝑈_𝜆). In fact, if 𝐵(𝑎,1/𝑛)⊄𝑓_𝜆⁻¹(𝑈_𝜆), then pick 𝑏∈𝐵(𝑎,1/𝑛)−𝑓_𝜆⁻¹(𝑈_𝜆). Note that 𝑓_𝜆⁻¹(𝑥)∩𝐵(𝑎,1/𝑛)≠∅, pick 𝑐∈𝑓_𝜆⁻¹(𝑥)∩𝐵(𝑎,1/𝑛), then 𝑑(𝑓_𝜆⁻¹(𝑥),𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆))≤𝑑(𝑐,𝑏)≤𝑑(𝑐,𝑎)+𝑑(𝑎,𝑏)<2/𝑛. It is a contradiction. So 𝐵(𝑎,1/𝑛)⊂𝑓_𝜆⁻¹(𝑈_𝜆), thus 𝑓_𝜆(𝐵(𝑎,1/𝑛))⊂𝑈_𝜆. Then st(𝑥,𝒫_𝜆,𝑛)⊂𝑈_𝜆. It implies that ⋃{st(𝑥,𝒫_𝜆,𝑛)∶𝜆∈Λwith𝑥∈𝑋_𝜆}⊂𝑈.
(2) ⇒ (1). For each 𝜆∈Λ, let 𝑥∈𝑈_𝜆 with 𝑈_𝜆 open in 𝑋_𝜆. We get that 𝑈_𝜆=𝑈∩𝑋_𝜆 with some 𝑈 open in 𝑋. Since ⋃{st(𝑥,𝒫_𝜆,𝑛)∶𝜆∈Λwith𝑥∈𝑋_𝜆}⊂𝑈 for some 𝑛∈ℕ, st(𝑥,𝒫_𝜆,𝑛)⊂𝑈_𝜆. It implies that {𝒫_𝜆,𝑛∶𝑛∈ℕ} is a point-star network for 𝑋_𝜆. Then the Ponomarev's system (𝑓_𝜆,𝑀_𝜆,𝑋_𝜆,{𝒫_𝜆,𝑛}) exists. Since each 𝒫_𝜆,𝑛 is countable, 𝑀_𝜆 is a separable metric space with metric 𝑑_𝜆 described as follows. For 𝑎=(𝛼_𝑛),𝑏=(𝛽_𝑛)∈𝑀_𝜆, if 𝑎=𝑏, then 𝑑_𝜆(𝑎,𝑏)=0, and if 𝑎≠𝑏, then 𝑑_𝜆(𝑎,𝑏)=1/(min{𝑛∈ℕ∶𝛼_𝑛≠𝛽_𝑛}).
Put 𝑀=⊕_𝜆∈Λ𝑀_𝜆 and define 𝑓∶𝑀→𝑋 by choosing 𝑓(𝑎)=𝑓_𝜆(𝑎) for every 𝑎∈𝑀_𝜆 with some 𝜆∈Λ. Then 𝑓 is a mapping and 𝑀 is a locally separable metric space with metric 𝑑 defined as follows. For 𝑎,𝑏∈𝑀, if 𝑎,𝑏∈𝑀_𝜆 for some 𝜆∈Λ, then 𝑑(𝑎,𝑏)=𝑑_𝜆(𝑎,𝑏), and otherwise, 𝑑(𝑎,𝑏)=1.
We will prove that 𝑓 is a 𝜋-mapping. Let 𝑥∈𝑈 with 𝑈 open in 𝑋, then ⋃{st(𝑥,𝒫_𝜆,𝑛)∶𝜆∈Λwith𝑥∈𝑋_𝜆}⊂𝑈 for some 𝑛∈ℕ. So, for each 𝜆∈Λ with 𝑥∈𝑋_𝜆, we get st(𝑥,𝒫_𝜆,𝑛)⊂𝑈_𝜆, where 𝑈_𝜆=𝑈∩𝑋_𝜆. It is implies that 𝑑_𝜆(𝑓_𝜆⁻¹(𝑥),𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆))≥1/𝑛. In fact, if 𝑎=(𝛼_𝑘)∈𝑀_𝜆 such that 𝑑_𝜆(𝑓_𝜆⁻¹(𝑥),𝑎)<1/𝑛, then there is 𝑏=(𝛽_𝑘)∈𝑓_𝜆⁻¹(𝑥) such that 𝑑_𝜆(𝑎,𝑏)<1/𝑛. So 𝛼_𝑘=𝛽_𝑘 if 𝑘≤𝑛. Note that 𝑥∈𝑃_𝛽𝑛⊂st(𝑥,𝒫_𝜆,𝑛)⊂𝑈_𝜆. Then 𝑓_𝜆(𝑎)∈𝑃_𝛼𝑛=𝑃_𝛽𝑛⊂st(𝑥,𝒫_𝜆,𝑛)⊂𝑈_𝜆. Hence 𝑎∈𝑓_𝜆⁻¹(𝑈_𝜆). It implies that 𝑑_𝜆(𝑓_𝜆⁻¹(𝑥),𝑎)≥1/𝑛 if 𝑎∈𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆). So 𝑑_𝜆(𝑓_𝜆⁻¹(𝑥),𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆))≥1/𝑛. Therefore𝑑(𝑓⁻¹(𝑥),𝑀−𝑓⁻¹(𝑈))=inf{𝑑(𝑎,𝑏)∶𝑎∈𝑓⁻¹(𝑥),𝑏∈𝑀−𝑓⁻¹(𝑈)}=min{1,inf{𝑑_𝜆(𝑎,𝑏)∶𝑎∈𝑓_𝜆⁻¹(𝑥),𝑏∈𝑀_𝜆−𝑓_𝜆⁻¹(𝑈_𝜆),𝜆∈Λ}}≥1/𝑛>0. (2.1)It implies that 𝑓 is a 𝜋-mapping.

Theorem 2.5. The following are equivalent for a space 𝑋.
(1)𝑋 is a compact-covering 𝜋-image of a locally separable metric space;(2)𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property satisfying that for each compact subset 𝐾 of 𝑋, there is a finite subset Λ_𝐾 of Λ such that 𝐾 has a finite compact cover {𝐾_𝜆∶𝜆∈Λ_𝐾}, and for each 𝜆∈Λ_𝐾 and 𝑛∈ℕ, 𝒫_𝜆,𝑛 is a 𝑘-cover for 𝐾_𝜆 in 𝑋_𝜆;(3) same as (2), but replace the prefix “𝑘-” by “𝑐𝑓𝑝-”.

Proof. (1) ⇒ (2). By using notations and arguments in the proof (1) ⇒ (2) of Proposition 2.4 again, 𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property. For each compact subset 𝐾 of 𝑋, since 𝑓 is compact-covering, 𝐾=𝑓(𝐿) for some compact subset 𝐿 of 𝑀. By compactness of 𝐿, 𝐿_𝜆=𝐿∩𝑀_𝜆 is compact and Λ_𝐾={𝜆∈Λ∶𝐿_𝜆≠∅} is finite. For each 𝜆∈Λ_𝐾, put 𝐾_𝜆=𝑓(𝐿_𝜆), then {𝐾_𝜆∶𝜆∈Λ_𝐾} is a finite compact cover for 𝐾. For each 𝑛∈ℕ, since ℬ_𝜆,𝑛 is a 𝑘-cover for 𝐿_𝜆 in 𝑀_𝜆, 𝒫_𝜆,𝑛 is a 𝑘-cover for 𝐾_𝜆 in 𝑋_𝜆 by Lemma 2.2.
(2) ⇒ (3). For each 𝜆∈Λ_𝐾 and 𝑛∈ℕ, since 𝒫_𝜆,𝑛 is countable, every member of 𝒫_𝜆,𝑛 can be chosen closed in 𝑋_𝜆. Thus, 𝒫_𝜆,𝑛 is a 𝑐𝑓𝑝-cover for 𝐾_𝜆 in 𝑋_𝜆 by Remark 1.2.
(3) ⇒ (1). By using notations and arguments in the proof (2) ⇒ (1) of Proposition 2.4 again, 𝑋 is a 𝜋-image of a locally separable metric space. It suffices to prove that the mapping 𝑓 is compact-covering.
For each compact subset 𝐾 of 𝑋, there is a finite subset Λ_𝐾 of Λ such that 𝐾 has a finite compact cover {𝐾_𝜆∶𝜆∈Λ_𝐾}, and for each 𝜆∈Λ_𝐾 and 𝑛∈ℕ, 𝒫_𝜆,𝑛 is a 𝑐𝑓𝑝-cover for 𝐾_𝜆 in 𝑋_𝜆. It follows from [13, Lemma 13] that 𝐾_𝜆=𝑓_𝜆(𝐿_𝜆) with some compact subset 𝐿_𝜆 of 𝑀_𝜆. Put 𝐿=⋃{𝐿_𝜆∶𝜆∈Λ_𝐾}, then 𝐿 is a compact subset of 𝑀 and 𝑓(𝐿)=𝐾. It implies that 𝑓 is compact-covering.

Theorem 2.6. The following are equivalent for a space 𝑋.
(1)𝑋 is a pseudo-sequence-covering 𝜋-image of a locally separable metric space;(2)𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property satisfying that for each convergent sequence 𝑆 of 𝑋, there is a finite subset Λ_𝑆 of Λ such that 𝑆 has a finite compact cover {𝑆_𝜆∶𝜆∈Λ_𝑆}, and for each 𝜆∈Λ_𝑆 and 𝑛∈ℕ, 𝒫_𝜆,𝑛 is a 𝑤𝑐𝑠-cover for 𝑆_𝜆 in 𝑋_𝜆;(3)same as (2), but replace the prefix “𝑤𝑐𝑠-” by “𝑐𝑠^∗-”.

Proof. Using Lemma 2.1, notations and arguments in the proof of Theorem 2.5, here “ pseudo-sequence-covering” and “convergent sequence” play the roles of “ compact-covering” and “compact subset”, respectively.

Theorem 2.7. The following are equivalent for a space 𝑋.
(1)𝑋 is a sequence-covering 𝜋-image of a locally separable metric space;(2)𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property satisfying that for each convergent sequence 𝑆 of 𝑋, there is 𝜆∈Λ such that 𝑆 is eventually in 𝑋_𝜆, and for each 𝑛∈ℕ, 𝒫_𝜆,𝑛 is a 𝑐𝑠-cover for 𝑆∩𝑋_𝜆 in 𝑋_𝜆.

Proof. (1) ⇒ (2). By using notations and arguments in the proof (1) ⇒ (2) of Proposition 2.4 again, 𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property. For each convergent sequence 𝑆 of 𝑋, since 𝑓 is sequence-covering, 𝑆=𝑓(𝐿) for some convergent sequence 𝐿 of 𝑀. Note that 𝐿 is eventually in some 𝑀_𝜆. Thus, 𝑆 is eventually in 𝑋_𝜆 with some 𝜆∈Λ. On the other hand, for each 𝑛∈ℕ, ℬ_𝜆,𝑛 is a 𝑐𝑠-cover for 𝐿∩𝑀_𝜆 in 𝑀_𝜆. It follows from Lemma 2.2 that 𝒫_𝜆,𝑛 is a 𝑐𝑠-cover for 𝑓(𝐿∩𝑀_𝜆) in 𝑋_𝜆. Then 𝒫_𝜆,𝑛 is a 𝑐𝑠-cover for 𝑆∩𝑋_𝜆 in 𝑋_𝜆.
(2) ⇒ (1). By using notations and arguments in the proof (2) ⇒ (1) of Proposition 2.4 again, 𝑋 is a 𝜋-image of a locally separable metric space. It suffices to prove that the mapping 𝑓 is sequence-covering.
For each convergent sequence 𝑆 of 𝑋, there is 𝜆∈Λ such that 𝑆 is eventually in 𝑋_𝜆, and for each 𝑛∈ℕ, 𝒫_𝜆,𝑛 is a 𝑐𝑠-cover for 𝑆∩𝑋_𝜆 in 𝑋_𝜆. Since 𝑆∩𝑋_𝜆 is a convergent sequence in 𝑋_𝜆, there is a convergent sequence 𝐿_𝜆 in 𝑀_𝜆 such that 𝑓_𝜆(𝐿_𝜆)=𝑆∩𝑋_𝜆 by Lemma 2.3. Note that 𝑆−𝑋_𝜆=𝑓(𝐹) with some finite subset 𝐹 of 𝑀. Put 𝐿=𝐹∪𝐿_𝜆, then 𝐿 is a convergent sequence in 𝑀 and 𝑓(𝐿)=𝑆. It implies that 𝑓 is sequence-covering.

Theorem 2.8. The following are equivalent for a space 𝑋.
(1)𝑋 is a subsequence-covering 𝜋-image of a locally separable metric space;(2)𝑋 is a sequentially-quotient 𝜋-image of a locally separable metric space;(3)𝑋 has a cover {𝑋_𝜆∶𝜆∈Λ} having 𝜋-property satisfying that for each convergent sequence 𝑆 of 𝑋, there is 𝜆∈Λ such that 𝒫_𝜆,𝑛 is a 𝑐𝑠-cover for some subsequence 𝑆_𝜆 of 𝑆 in 𝑋_𝜆 for each 𝑛∈ℕ;(4)same as (3), but replace the prefix “𝑐𝑠-” by “𝑤𝑐𝑠-”, or “𝑐𝑠^∗-”.

Proof. (1) ⇒ (2). By [4, Proposition 2.1].
(2) ⇒ (3). For each convergent sequence 𝑆 of 𝑋, since 𝑓 is sequentially-quotient, there is a convergent sequence 𝐿 of 𝑀 such that 𝑓(𝐿) is a subsequence of 𝑆. By the proof of (1) ⇒ (2) in Theorem 2.7, where 𝑓(𝐿) plays the role of 𝑆 in the argument, we get (3).
(3) ⇒ (4). By Lemma 2.1, “𝑤𝑐𝑠-” and “𝑐𝑠^∗-” are equivalent. It is routine when “𝑐𝑠-” is replaced by “𝑐𝑠^∗-”.
(4) ⇒ (1). It suffices to assume that 𝒫_𝜆,𝑛 is a 𝑐𝑠^∗-cover for a subsequence 𝑇 of 𝑆 in 𝑋_𝜆. By the proof of (3) ⇒ (1) in Theorem 2.5, where 𝑇 plays the role of compact subset 𝐾 in the argument, we get (1).

Based on above results, it is easy to get characterizations for quotient 𝜋-images of locally separable metric spaces as follows.

Corollary 2.9. The following are equivalent for a space 𝑋.
(1)𝑋 is a compact-covering quotient 𝜋-image of a locally separable metric space;(2)𝑋 is a sequential space satisfying (2), or (3) in Theorem 2.5.

Corollary 2.10. The following are equivalent for a space 𝑋.
(1)𝑋 is a pseudo-sequence-covering quotient 𝜋-image of a locally separable metric space;(2)𝑋 is a sequential space satisfying (2), or (3) in Theorem 2.6.

Corollary 2.11. The following are equivalent for a space 𝑋.
(1)𝑋 is a sequence-covering quotient 𝜋-image of a locally separable metric space;(2)𝑋 is a sequential space satisfying (2) in Theorem 2.7.

Corollary 2.12. The following are equivalent for a space 𝑋.
(1)𝑋 is a quotient 𝜋-image of a locally separable metric space;(2)𝑋 is a sequential space satisfying (3), or (4) in Theorem 2.8.

Remark 2.13. “Quotient” and “sequential” in above corollaries can be replaced by “pseudo-open” and “Fréchet”, respectively.

In [6], the authors raised a question whether pseudo-sequence-covering quotient 𝜋-images of metric spaces and quotient 𝜋-images of metric spaces are equivalent. Similar to this question we have the following.

Question 2.14. Are pseudo-sequence-covering quotient 𝜋-images of locally separable metric spaces and quotient 𝜋-images of locally separable metric spaces equivalent?

Acknowledgments

The authors would like to thank the referee for his/her valuable comments. This is supported in part by the National Natural Science Foundation of Vietnam (Grant no. 1 008 06).