On feebly compact shift-continuous topologies on the semilattice $\exp_n\lambda$

TL;DR

This paper investigates feebly compact shift-continuous topologies on a specific semilattice, establishing their equivalence with several compactness conditions and $H$-closedness in the context of $T_1$-topologies.

Contribution

It characterizes the equivalence of various compactness properties and $H$-closedness for shift-continuous $T_1$-topologies on the semilattice $ ext{exp}_n ext{lambda}$.

Findings

Feebly compact topologies are equivalent to countably pracompact and $d$-feebly compact topologies.

Such topologies are also $H$-closed spaces.

The results unify several compactness conditions under shift-continuous $T_1$-topologies.

Abstract

We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice and prove that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$~$\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space.

Equations

$$\uparrow e=\left\{f\in S:e⩽f\right\}.$$

$${\exp }_n\lambda =\left\{A\subseteq \lambda :|A|⩽n\right\}.$$

$$a_j=\left\{x_{n(j-1)+1},x_{n(j-1)+2},\dots ,x_{nj}\right\}.$$

$${\exp }_n\lambda \setminus {\mathrm{cl}}_{{\exp }_n\lambda }(U(0))\subseteq \uparrow x_1\cup \cdots \cup \uparrow x_m.$$

$${\exp }_n\lambda \setminus {\mathrm{cl}}_{{\exp }_n\lambda }(U(0))⊈\uparrow x_1\cup \cdots \cup \uparrow x_m$$

$$\left({\exp }_n\lambda \setminus {\mathrm{cl}}_{{\exp }_n\lambda }(U(0))\right)\cap \left(\uparrow y_2\setminus \uparrow y_1\right)\ne \varnothing .$$

$$m_2\in \left({\exp }_n\lambda \setminus {\mathrm{cl}}_{{\exp }_n\lambda }(U(0))\right)\cap \left(\uparrow y_2\setminus \uparrow y_1\right).$$

$$\left\{\left\{m_i\right\}:i=1,2,3,\dots \right\}$$

$$\uparrow x\setminus {\mathrm{cl}}_{{\exp }_n\lambda }(U(x))\subseteq \uparrow x_1\cup \cdots \cup \uparrow x_m.$$

$${\exp }_k\lambda \cap U(x)\subseteq \uparrow x_1\cup \cdots \cup \uparrow x_m.$$

$$W(x)=U(x)\setminus \left(\uparrow x_1\cup \cdots \cup \uparrow x_m\right)$$

$${\mathcal{U}}_1=\left\{U\in {\mathcal{U}}_0:U(x_1)\cap U=\varnothing \right\}.$$

$${\mathcal{U}}_2=\left\{U\in {\mathcal{U}}_1:U(x_2)\cap U=\varnothing \right\}.$$

$${\mathcal{U}}_k=\left\{U\in {\mathcal{U}}_{k-1}:U(x_1)\cap U=\dots =U(x_k)\cap U=\varnothing \right\}.$$

$$\mathcal{U}=\left\{U_i:i=1,2,3,\dots \right\}and\mathcal{V}=\left\{V_i:i=1,2,3,\dots \right\}$$

$$\overline{\mathcal{V}}=\left\{{\mathrm{cl}}_X(V_i):i=1,2,3,\dots \right\}$$

$$(a,b)\cdot(c,d)=\left\{\begin{array}[]{cl}(a,d),&\hbox{ if~{} }b=c;\\ 0,&\hbox{ if~{} }b\neq c,\end{array}\right.$$

$$P_{\lambda }=⟨{\left\{p_i\right\}}_{i\in \lambda },{\left\{p_i^{-1}\right\}}_{i\in \lambda }\mid p_i{p}_i^{-1}=1,p_i{p}_j^{-1}=0fori\ne j⟩$$

Full text

Dedicated to the memory of Professor Vitaly Sushchanskyy

We shall follow the terminology of [6, 8, 9, 13]. If $X$ is a topological space and $A\subseteq X$, then by $\operatorname{cl}_{X}(A)$ and $\operatorname{int}_{X}(A)$ we denote the closure and the interior of $A$ in $X$, respectively. By $\omega$ we denote the first infinite cardinal and by $\mathbb{N}$ the set of positive integers.

A subset $A$ of a topological space $X$ is called regular open if $\operatorname{int}_{X}(\operatorname{cl}_{X}(A))=A$.

We recall that a topological space $X$ is said to be

• •

quasiregular if for any non-empty open set $U\subset X$ there exists a non-empty open set $V\subset U$ such that $\operatorname{cl}_{X}(V)\subseteq U$;

• •

semiregular if $X$ has a base consisting of regular open subsets;

• •

compact if each open cover of $X$ has a finite subcover;

• •

countably compact if each open countable cover of $X$ has a finite subcover;

• •

countably compact at a subset $A\subseteq X$ if every infinite subset $B\subseteq A$ has an accumulation point $x$ in $X$;

• •

countably pracompact if there exists a dense subset $A$ in $X$ such that $X$ is countably compact at $A$;

• •

feebly compact (or lightly compact) if each locally finite open cover of $X$ is finite [3];

• •

$d$-feebly compact (or DFCC) if every discrete family of open subsets in $X$ is finite (see [12]);

• •

pseudocompact if $X$ is Tychonoff and each continuous real-valued function on $X$ is bounded.

According to Theorem 3.10.22 of [8], a Tychonoff topological space $X$ is feebly compact if and only if $X$ is pseudocompact. Also, a Hausdorff topological space $X$ is feebly compact if and only if every locally finite family of non-empty open subsets of $X$ is finite [3]. Every compact space and every sequentially compact space are countably compact, every countably compact space is countably pracompact, and every countably pracompact space is feebly compact (see [2]), and every $H$-closed space is feebly compact too (see [10]). Also, it is obvious that every feebly compact space is $d$-feebly compact.

A semilattice is a commutative semigroup of idempotents. On a semilattice $S$ there exists a natural partial order: $e\leqslant f$ if and only if $ef=fe=e$. For any element $e$ of a semilattice $S$ we put

[TABLE]

A topological (semitopological) semilattice is a topological space together with a continuous (separately continuous) semilattice operation. If $S$ is a semilattice and $\tau$ is a topology on $S$ such that $(S,\tau)$ is a topological semilattice, then we shall call $\tau$ a semilattice topology on $S$, and if $\tau$ is a topology on $S$ such that $(S,\tau)$ is a semitopological semilattice, then we shall call $\tau$ a shift-continuous topology on $S$.

For an arbitrary positive integer $n$ and an arbitrary non-zero cardinal $\lambda$ we put

[TABLE]

It is obvious that for any positive integer $n$ and any non-zero cardinal $\lambda$ the set $\exp_{n}\lambda$ with the binary operation $\cap$ is a semilattice. Later in this paper by $\exp_{n}\lambda$ we shall denote the semilattice $\left(\exp_{n}\lambda,\cap\right)$.

This paper is a continuation of [11] where we study feebly compact topologies $\tau$ on the semilattice $\exp_{n}\lambda$ such that $\left(\exp_{n}\lambda,\tau\right)$ is a semitopological semilattice. Therein, all compact semilattice $T_{1}$-topologies on $\exp_{n}\lambda$ were described. In [11] it was proved that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every $T_{1}$-semitopological countably compact semilattice $\left(\exp_{n}\lambda,\tau\right)$ is a compact topological semilattice. Also, there we construct a countably pracompact $H$-closed quasiregular non-semiregular topology $\tau_{\operatorname{\textsf{fc}}}^{2}$ such that $\left(\exp_{2}\lambda,\tau_{\operatorname{\textsf{fc}}}^{2}\right)$ is a semitopological semilattice with the discontinuous semilattice operation and show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ a semiregular feebly compact semitopological semilattice $\exp_{n}\lambda$ is a compact topological semilattice.

In this paper we show that for any shift-continuous $T_{1}$-topology $\tau$ on $\exp_{n}\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\exp_{n}\lambda,\tau\right)$ is an $H$-closed space.

The proof of the following lemma is similar to Lemma 4.5 of [5] or Proposition 1 from [1].

We observe that by Proposition 1 from [11] for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every shift-continuous $T_{1}$-topology $\tau$ on $\exp_{n}\lambda$ is functionally Hausdorff and quasiregular, and hence it is Hausdorff.

It is obvious that the subsemilattice ${\uparrow}y$ of $\exp_{n}\lambda$ is algebraically isomorphic to the semilattice $\exp_{k}\lambda$ for some positive integer $k\leqslant n$. This and above arguments imply that without loss of generality we may assume that $y$ is the isolated zero of the $d$-feebly compact semitopological semilattice $\left(\exp_{n}\lambda,\tau\right)$.

Hence we assume that $\tau$ is a $d$-feebly compact shift-continuous topology on $\exp_{n}\lambda$ such that the zero [math] of $\exp_{n}\lambda$ is an isolated point of $\left(\exp_{n}\lambda,\tau\right)$. Next we fix an arbitrary infinite sequence $\left\{x_{i}\right\}_{i\in\mathbb{N}}$ of distinct elements of cardinal $\lambda$. For every positive integer $j$ we put

[TABLE]

Then $a_{j}\in\exp_{n}\lambda$ and moreover $a_{j}$ is a greatest element of the semilattice $\exp_{n}\lambda$ for each positive integer $j$. Also, the definition of the semilattice $\exp_{n}\lambda$ implies that for every non-zero element $a$ of $\exp_{n}\lambda$ there exists at most one element $a_{j}$ such that $a_{j}\in{\uparrow}a$. Then for every positive integer $j$ by Proposition 1$(iii)$ of [11], $a_{j}$ is an isolated point of $\left(\exp_{n}\lambda,\tau\right)$, and hence the above arguments imply that $\left\{a_{1},a_{2},\ldots,a_{j},\ldots\right\}$ is an infinite discrete family of open subset in the space $\left(\exp_{n}\lambda,\tau\right)$. This contradicts the $d$-feeble compactness of the semitopological semilattice $\left(\exp_{n}\lambda,\tau\right)$. The obtained contradiction implies the statement of our proposition. ∎

The following example show that the converse statement to Proposition 1 is not true in the case of topological semilattices.

• $(i)$

all non-zero elements of the semilattice $\exp_{1}\lambda$ are isolated points in $\left(\exp_{1}\lambda,\tau_{\operatorname{\textsf{dm}}}\right)$;   and

• $(ii)$

the family $\mathscr{B}_{\operatorname{\textsf{dm}}}=\left\{U_{B}=\{0\}\cup\pi(B)\colon B\subseteq A\hbox{~{}and~{}}A\setminus B\hbox{~{}is finite}\right\}$ is the base of the topology $\tau_{\operatorname{\textsf{dm}}}$ at zero [math] of $\exp_{1}\lambda$.

Simple verifications show that $\tau_{\operatorname{\textsf{dm}}}$ is a Hausdorff locally compact semilattice topology on $\exp_{1}\lambda$ which is not compact and hence by Corollary 8 of [11] it is not feebly compact.

Fix an arbitrary subset $A$ in $\lambda$ such that $\left|\lambda\setminus A\right|\geqslant\omega$. For every zero element $y\in\exp_{n}\lambda\setminus{\uparrow}x$ we assume that the base $\mathscr{B}_{\operatorname{\textsf{dm}}}^{n}(y)$ of the topology $\tau_{\operatorname{\textsf{dm}}}^{n}$ at the point $y$ coincides with the base of the topology $\tau_{\operatorname{\textsf{c}}}^{n}$ at $y$, and assume that ${\uparrow}x$ is an open-and-closed subset and the topology on ${\uparrow}x$ is generated by the map $h\colon\left(\exp_{2}\lambda,\tau_{\operatorname{\textsf{fc}}}^{2}\right)\to{\uparrow}x$. We observe that $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$ is a Hausdorff locally compact topological space, because it is the topological sum of a Hausdorff locally compact space ${\uparrow}x$ (which is homeomorphic to the Hausdorff locally compact space $\left(\exp_{1}\lambda,\tau_{\operatorname{\textsf{dm}}}\right)$ from Example 1) and an open-and-closed subspace $\exp_{n}\lambda\setminus{\uparrow}x$ of $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{c}}}^{n}\right)$. It is obvious that the set $\exp_{n}\lambda\setminus\exp_{n-1}\lambda$ is dense in $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$. Also, since ${\uparrow}x$ is an open-and-closed subsemilattice with zero $x$ of $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$, the continuity of the semilattice operations in $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$ and $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{c}}}^{n}\right)$ and the property that the topology $\tau_{\operatorname{\textsf{dm}}}^{n}$ is more stronger them $\tau_{\operatorname{\textsf{c}}}^{n}$, imply that $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$ is a topological semilattice. Moreover, the space $\left(\exp_{n}\lambda,\tau_{\operatorname{\textsf{dm}}}^{n}\right)$ is not $d$-feebly compact, because it contains an open-and-closed non-$d$-feebly compact subspace ${\uparrow}x$.

Arguments presented in the proof of Proposition 1 and Proposition 1$(iii)$ of [11] imply the following corollary.

The following proposition gives an amazing property of the system of neighbourhoodd of zero in a $T_{1}$-feebly compact semitopological semilattice $\exp_{n}\lambda$.

[TABLE]

[TABLE]

for any finitely many $x_{1},\ldots,x_{m}\in\lambda$.

We fix an arbitrary $y_{1}\in\lambda$ such that $\left(\exp_{n}\lambda\setminus\operatorname{cl}_{\exp_{n}\lambda}(U(0))\right)\cap{\uparrow}y_{1}\neq\varnothing$. By Proposition 1$(iii)$ of [11] the set ${\uparrow}y_{1}$ is open in $\left(\exp_{n}\lambda,\tau\right)$ and hence the set $\left(\exp_{n}\lambda\setminus\operatorname{cl}_{\exp_{n}\lambda}(U(0))\right)\cap{\uparrow}y_{1}$ is open in $\left(\exp_{n}\lambda,\tau\right)$ too. Then by Proposition 1 there exists an isolated point $m_{1}\in\exp_{n}\lambda\setminus\exp_{n-1}\lambda$ in $\left(\exp_{n}\lambda,\tau\right)$ such that $m_{1}\in\left(\exp_{n}\lambda\setminus\operatorname{cl}_{\exp_{n}\lambda}(U(0))\right)\cap{\uparrow}y_{1}$. Now, by the assumption there exists $y_{2}\in\lambda$ such that

[TABLE]

Again, since by Proposition 1$(iii)$ of [11] both sets ${\uparrow}y_{1}$ and ${\uparrow}y_{2}$ are open-and-closed in $\left(\exp_{n}\lambda,\tau\right)$, Proposition 1 implies that there exists an isolated point $m_{2}\in\exp_{n}\lambda\setminus\exp_{n-1}\lambda$ in $\left(\exp_{n}\lambda,\tau\right)$ such that

[TABLE]

Hence by induction we can construct a sequence $\left\{y_{i}\colon i=1,2,3,\ldots\right\}$ of distinct points of $\lambda$ and a sequence of isolated points $\left\{m_{i}\colon i=1,2,3,\ldots\right\}\subset\exp_{n}\lambda\setminus\exp_{n-1}\lambda$ in $\left(\exp_{n}\lambda,\tau\right)$ such that for any positive integer $k$ the following conditions hold:

• $(i)$

$\left(\exp_{n}\lambda\setminus\operatorname{cl}_{\exp_{n}\lambda}(U(0))\right)\cap\left({\uparrow}y_{k}\setminus\left({\uparrow}y_{1}\cup\cdots\cup{\uparrow}y_{k-1}\right)\right)\neq\varnothing$;   and

• $(ii)$

$m_{k}\in\left(\exp_{n}\lambda\setminus\operatorname{cl}_{\exp_{n}\lambda}(U(0))\right)\cap\left({\uparrow}y_{k}\setminus\left({\uparrow}y_{1}\cup\cdots\cup{\uparrow}y_{k-1}\right)\right)$.

Then similar arguments as in the proof of Proposition 1 imply that the following family

[TABLE]

is infinite and locally finite, which contradicts the feeble compactness of $\left(\exp_{n}\lambda,\tau\right)$. The obtained contradiction implies the statement of the proposition. ∎

Proposition 1$(iii)$ of [11] implies that for any element $x\in\exp_{n}\lambda$ the set ${\uparrow}x$ is open-and-closed in a $T_{1}$-semitopological semilattice $\left(\exp_{n}\lambda,\tau\right)$ and hence by Theorem 14 from [3] we have that for any $x\in\exp_{n}\lambda$ the space ${\uparrow}x$ is feebly compact in a feebly compact $T_{1}$-semitopological semilattice $\left(\exp_{n}\lambda,\tau\right)$. Hence Proposition 2 implies the following proposition.

[TABLE]

The main results of this paper is the following theorem.

• $(i)$

$\tau$* is countably pracompact;*

• $(ii)$

$\tau$* is feebly compact;*

• $(iii)$

$\tau$* is $d$-feebly compact;*

• $(iv)$

the space $\left(\exp_{n}\lambda,\tau\right)$ is $H$-closed.

Implication $(iv)\Rightarrow(ii)$ follows from Proposition 4 of [10].

$(ii)\Rightarrow(iv)$ We shall prove this implication by induction.

By Corollary 2 from [11] every feebly compact $T_{1}$-topology $\tau$ on the semilattice $\exp_{1}\lambda$ such that $\left(\exp_{1}\lambda,\tau\right)$ is a semitopological semilattice, is compact, and hence $\left(\exp_{1}\lambda,\tau\right)$ is an $H$-closed topological space.

Next we shall show that if our statements holds for all positive integers $j<k\leqslant n$ then it holds for $j=k$. Suppose that a feebly compact $T_{1}$-semitopological semilattice $\left(\exp_{k}\lambda,\tau\right)$ is a subspace of Hausdorff topological space $X$. Fix an arbitrary point $x\in X$ and an arbitrary open neighbourhood $V(x)$ of $x$ in $X$. Since $X$ is Hausdorff, there exist disjoint open neighbourhoods $U(x)\subseteq V(x)$ and $U(0)$ of $x$ and zero [math] of the semilattice $\exp_{k}\lambda$ in $X$, respectively. Then $\operatorname{cl}_{X}(U(0))\cap U(x)=\varnothing$ and hence by Proposition 2 there exists finitely many $x_{1},\ldots,x_{m}\in\lambda$ such that

[TABLE]

But for any $x\in\lambda$ the subsemilattice ${\uparrow}x$ of $\exp_{k}\lambda$ is algebraically isomorphic to the semilattice $\exp_{k-1}\lambda$. Then by Proposition 1$(iii)$ of [11] and Theorem 14 from [3], ${\uparrow}x$ is a feebly compact $T_{1}$-semitopological semilattice, and the assumption of our induction implies that ${\uparrow}x_{1},\cdots,{\uparrow}x_{m}$ are closed subsets of $X$. This implies that

[TABLE]

is an open neighbourhood of $x$ in $X$ such that $W(x)\cap\exp_{k}\lambda=\varnothing$. Thus, $\left(\exp_{k}\lambda,\tau\right)$ is an $H$-closed space. This completes the proof of the requested implication. ∎

The following theorem gives a sufficient condition when a $d$-feebly compact space is feebly compact.

By induction we shall construct an infinite discrete family of non-empty open subsets of $X$.

Fix an arbitrary $U_{1}\in\mathscr{U}_{0}$ and an arbitrary point $x_{1}\in U_{1}$. Since the family $\mathscr{U}_{0}$ is locally finite there exists an open neighbourhood $U(x_{1})\subseteq U_{1}$ of the point $x_{1}$ in $X$ such that $U(x_{1})$ intersects finitely many elements of $\mathscr{U}_{0}$. Also, the quasiregularity of $X$ implies that there exists a non-empty open subset $V_{1}\subseteq U(x_{1})$ such that $\operatorname{cl}_{X}(V_{1})\subseteq U(x_{1})$. Put

[TABLE]

Since the family $\mathscr{U}_{0}$ is locally finite and infinite, so is $\mathscr{U}_{1}$. Fix an arbitrary $U_{2}\in\mathscr{U}_{1}$ and an arbitrary point $x_{2}\in U_{2}$. Since the family $\mathscr{U}_{1}$ is locally finite, there exists an open neighbourhood $U(x_{2})\subseteq U_{2}$ of the point $x_{2}$ in $X$ such that $U(x_{2})$ intersects finitely many elements of $\mathscr{U}_{1}$. Since $X$ is quasiregular, there exists a non-empty open subset $V_{2}\subseteq U(x_{2})$ such that $\operatorname{cl}_{X}(V_{2})\subseteq U(x_{2})$. Our construction implies that the closed sets $\operatorname{cl}_{X}(V_{1})$ and $\operatorname{cl}_{X}(V_{2})$ are disjoint and hence so are $V_{1}$ and $V_{2}$. Next we put

[TABLE]

Also, we observe that it is obvious that $U(x_{1})\cap U=\varnothing$ for each $U\in\mathscr{U}_{1}$.

Suppose for some positive integer $k>1$ we construct:

• $(a)$

a sequence of infinite locally finite subfamilies $\mathscr{U}_{1},\ldots,\mathscr{U}_{k-1}$ in $\mathscr{U}_{0}$ of non-empty open subsets in the space $X$;

• $(b)$

a sequence of open subsets $U_{1},\ldots,U_{k}$ in $X$;

• $(c)$

a sequence of points $x_{1},\ldots,x_{k}$ in $X$ and a sequence of their corresponding open neighbourhoods $U(x_{1}),\ldots,U(x_{k})$ in $X$;

• $(d)$

a sequence of disjoint non-empty subsets $V_{1},\ldots,V_{k}$ in $X$

such that the following conditions hold:

• $(i)$

$\mathscr{U}_{i}$ is a proper subfamily of $\mathscr{U}_{i-1}$;

• $(ii)$

$U_{i}\in\mathscr{U}_{i-1}$ and $U_{i}\cap U=\varnothing$ for each $U\in\mathscr{U}_{j}$ with $i\leqslant j\leqslant k$;

• $(iii)$

$x_{i}\in U_{i}$ and $U(x_{i})\subseteq U_{i}$;

• $(iv)$

$V_{i}$ is an open subset of $U_{i}$ with $\operatorname{cl}_{X}(V_{i})\subseteq U(x_{i})$,

for all $i=1,\ldots,k$, and

• $(v)$

$\operatorname{cl}_{X}(V_{1}),\ldots,\operatorname{cl}_{X}(V_{k})$ are disjoint.

Next we put

[TABLE]

Since the family $\mathscr{U}_{k-1}$ is infinite and locally finite, there exists a subfamily $\mathscr{U}_{k}$ in $\mathscr{U}_{k-1}$ which is infinite and locally finite. Fix an arbitrary $U_{k+1}\in\mathscr{U}_{k}$ and an arbitrary point $x_{k+1}\in U_{k+1}$. Since the family $\mathscr{U}_{k}$ is locally finite, there exists an open neighbourhood $U(x_{k+1})\subseteq U_{k+1}$ of the point $x_{k+1}$ in $X$ such that $U(x_{k+1})$ intersects finitely many elements of $\mathscr{U}_{k}$. Since the space $X$ is quasiregular, there exists a non-empty open subset $V_{k+1}\subseteq U(x_{k+1})$ such that $\operatorname{cl}_{X}(V_{k+1})\subseteq U(x_{k+1})$. Simple verifications show that the conditions $(i)-(iv)$ hold in the case of the positive integer $k+1$.

Hence by induction we construct the following two infinite countable families of open non-empty subsets of $X$:

[TABLE]

such that $\operatorname{cl}_{X}(V_{i})\subseteq U_{i}$ for each positive integer $i$. Since $\mathscr{U}$ is a subfamily of $\mathscr{U}_{0}$ and $\mathscr{U}_{0}$ is locally finite in $X$, $\mathscr{U}$ is locally finite in $X$ as well. Also, above arguments imply that $\mathscr{V}$ and

[TABLE]

are locally finite families in $X$ too.

Next we shall show that the family $\mathscr{V}$ is discrete in $X$. Indeed, since the family $\overline{\mathscr{V}}$ is locally finite in $X$, by Theorem 1.1.11 of [8] the union $\bigcup\overline{\mathscr{V}}$ is a closed subset of $X$, and hence any point $x\in X\setminus\bigcup\overline{\mathscr{V}}$ has an open neighbourhood $O(x)=X\setminus\bigcup\overline{\mathscr{V}}$ which does not intersect the elements of the family $\mathscr{V}$. If $x\in\operatorname{cl}_{X}(V_{i})$ for some positive integer $i$, then our construction implies that $U(x_{i})$ is an open neighbourhood of $x$ which intersects only the set $V_{i}\in\mathscr{V}$. Hence $X$ has an infinite discrete family $\mathscr{V}$ of non-empty open subsets in $X$, which contradicts the assumption that the space $X$ is $d$-feebly compact. The obtained contradiction implies the statement of the theorem. ∎

We finish this note by some simple remarks about dense embedding of an infinite semigroup of matrix units and a polycyclic monoid into $d$-feebly compact topological semigroups which follow from the results of the paper [5].

Let $\lambda$ be a non-zero cardinal. On the set $B_{\lambda}=(\lambda\times\lambda)\cup\{0\}$, where $0\notin\lambda\times\lambda$, we define the semigroup operation “$\,\cdot\,$” as follows

[TABLE]

and $(a,b)\cdot 0=0\cdot(a,b)=0\cdot 0=0$ for $a,b,c,d\in\lambda$. The semigroup $B_{\lambda}$ is called the semigroup of $\lambda{\times}\lambda$-matrix units (see [7]).

The bicyclic monoid ${\mathscr{C}}(p,q)$ is the semigroup with the identity $1$ generated by two elements $p$ and $q$ subjected only to the condition $pq=1$ [7]. For a non-zero cardinal $\lambda$, the polycyclic monoid $P_{\lambda}$ on $\lambda$ generators is the semigroup with zero given by the presentation:

[TABLE]

(see [5]). It is obvious that in the case when $\lambda=1$ the semigroup $P_{1}$ is isomorphic to the bicyclic semigroup with adjoined zero.

By Theorem 4.4 from [5] for every infinite cardinal $\lambda$ the semigroup of $\lambda{\times}\lambda$-matrix units $B_{\lambda}$ does not densely embed into a Hausdorff feebly compact topological semigroup, and by Theorem 4.5 from [5] for arbitrary cardinal $\lambda\geqslant 2$ there exists no Hausdorff feebly compact topological semigroup which contains the $\lambda$-polycyclic monoid $P_{\lambda}$ as a dense subsemigroup. These theorems and Lemma 1 imply the following two corollaries.

The proof of the following corollary is similar to Theorem 5.1(5) from [4].

Acknowledgements

We acknowledge Alex Ravsky and the referee for their comments and suggestions.