Cardinal invariants of paratopological groups

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Cardinal invariants of paratopological groups

Iván Sánchez

Universidad Autónoma Metropolitana, México, D.F.

[email protected] São Sebastião, Brazil

Brazilian Conference

on General Topology and Set Theory In honour of Ofelia T. Alas

August 12–16, 2013

Iván Sánchez (UAM-I) Cardinal invariants of paratopological groups August 12–16, 2013 1 / 11

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1 Some definitions

2 A question of Arhangel’skii and Bella

3 The Hausdorff case

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Some definitions

Aparatopological groupis a group endowed with a topology for which multiplication in the group is continuous.

Now, consider a topological spaceX. LetU be a collection of subsets ofX. Thestar ofU with respect toA⊆X, denoted byst(U,A), is the setS

{U∈ U :U∩A6=∅}. WhenA={x}, we simply writest(U,x). We putst¹(U,x) =st(U,x)and recursively define

stⁿ⁺¹(U,x) =st(U,stⁿ(U,x)).

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Some definitions

Aparatopological group

is a group endowed with a topology for which multiplication in the group is continuous.

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Some definitions

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Some definitions

Now, consider a topological spaceX. LetU be a collection of subsets ofX.

Thestar ofU with respect toA⊆X, denoted byst(U,A), is the setS

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Some definitions

{U∈ U :U∩A6=∅}.

WhenA={x}, we simply writest(U,x). We putst¹(U,x) =st(U,x)and recursively define

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Some definitions

{U∈ U :U∩A6=∅}. WhenA={x}, we simply writest(U,x).

We putst¹(U,x) =st(U,x)and recursively define stⁿ⁺¹(U,x) =st(U,stⁿ(U,x)).

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Some definitions

{U∈ U :U∩A6=∅}. WhenA={x}, we simply writest(U,x).

We putst¹(U,x) =st(U,x)and recursively define stⁿ⁺¹(U,x) =st(U,stⁿ(U,x)).

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Some definitions

The rank of the diagonal

Letnbe a positive integer. We say that a spaceX has aG_δ-diagonal of ranknif there exists a countable collection{U_k :k ∈ω}of open covers ofX such thatT{stⁿ(U_k,x) :k ∈ω}={x}for eachx ∈X. If a space has aG_δ-diagonal of any possible rank, then we say that it has a G_δ-diagonal of infinite rank.

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Some definitions

The rank of the diagonal

Letnbe a positive integer. We say that a spaceX has aG_δ-diagonal of rankn

if there exists a countable collection{U_k :k ∈ω}of open covers ofX such thatT{stⁿ(U_k,x) :k ∈ω}={x}for eachx ∈X. If a space has aG_δ-diagonal of any possible rank, then we say that it has a G_δ-diagonal of infinite rank.

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Some definitions

The rank of the diagonal

Letnbe a positive integer. We say that a spaceX has aG_δ-diagonal of ranknif there exists a countable collection{U_k :k ∈ω}of open covers ofX such thatT{stⁿ(U_k,x) :k ∈ω}={x}for eachx ∈X.

If a space has aG_δ-diagonal of any possible rank, then we say that it has a G_δ-diagonal of infinite rank.

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Some definitions

The rank of the diagonal

Letnbe a positive integer. We say that a spaceX has aG_δ-diagonal of ranknif there exists a countable collection{U_k :k ∈ω}of open covers ofX such thatT{stⁿ(U_k,x) :k ∈ω}={x}for eachx ∈X. If a space has aG_δ-diagonal of any possible rank, then we say that it has a G_δ-diagonal of infinite rank.

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A question of Arhangel’skii and Bella

Theorem 1 (Arhangel’skii and A. Bella, 2007).

Every Hausdorfffirst countableparatopological group has a G_δ-diagonal of infinite rank.

In topological groups: countableπ-character ⇒countable character. In paratopological groups, we have:

Example 2 (Arhangel’skii and Burke, 2006).

There exists a Tychonoff paratopological groupGwith a countable π-base such that the spaceGis not first countable.

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In topological groups: countableπ-character ⇒countable character. In paratopological groups, we have:

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In topological groups: countableπ-character⇒countable character. In paratopological groups, we have:

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In topological groups: countableπ-character⇒countable character. In paratopological groups, we have:

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A question of Arhangel’skii and Bella

Problem 1 (Arhangel’skii and A. Bella, 2007).

Does a Hausdorff (regular, Tychonoff) paratopological group of countableπ-characterhave aG_δ-diagonal of infinite rank?

The next theorem resolvesProblem 1forregularparatopological groups.

Theorem 3.

Every regular paratopological group of countableπ-character has a G_δ-diagonal of infinite rank.

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A question of Arhangel’skii and Bella

Theorem 3.

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A question of Arhangel’skii and Bella

Theorem 3.

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The Hausdorff case

Regularization of paratopological groups

Let(X, τ)a topological space. The family{IntU :U ∈τ}is a base for a topologyτr onX. The spaceXr = (X, τr)is called the

semiregularizationofX (Stone-Kat ˇetov). Clearly, the topologyτr is weaker thanτ.

Theorem 4 (Ravsky, 2003).

IfGis a Hausdorff paratopological group, then the semiregularization G_r ofGis a regular paratopological group.

The previous fact permits us to callG_r theregularizationofG.

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The Hausdorff case

Regularization of paratopological groups

Let(X, τ)a topological space. The family{IntU :U ∈τ}is a base for a topologyτr onX.

The spaceXr = (X, τr)is called the

semiregularizationofX (Stone-Kat ˇetov). Clearly, the topologyτr is weaker thanτ.

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The Hausdorff case

Regularization of paratopological groups

semiregularizationofX (Stone-Kat ˇetov). Clearly, the topologyτ_r is weaker thanτ.

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The Hausdorff case

Regularization of paratopological groups

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The Hausdorff case

Regularization of paratopological groups

The previous fact permits us to callGr theregularizationofG.

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The Hausdorff case

The regularization of subgroups

LetHbe a subgroup of a paratopological group(G, τ). Putσ =τ|_H.

In general, the topologiesτr|_H andσr are different.

Lemma 5.

IfH is a dense subgroup of a paratopological group(G, τ)and σ =τ|_H, thenτr|_H =σr.

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The Hausdorff case

The regularization of subgroups

LetHbe a subgroup of a paratopological group(G, τ). Putσ =τ|_H. In general, the topologiesτr|_H andσr are different.

Lemma 5.

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The Hausdorff case

The regularization of subgroups

LetHbe a subgroup of a paratopological group(G, τ). Putσ =τ|_H. In general, the topologiesτr|_H andσr are different.

Lemma 5.

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The Hausdorff case

Now, we resolveProblem 1. Theorem 6.

LetHbe a dense subgroup of a Hausdorff paratopological group (G, τ)such thatH has countableπ-character. Then(G, τ)has a G_δ-diagonal of infinite rank.

Proof.

Putσ=τ|_H. Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω. Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character. Theorem 3implies thatGr has aG_δ-diagonal of infinite rank. Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Now, we resolveProblem 1.

Theorem 6.

Proof.

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The Hausdorff case

Theorem 6.

Proof.

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The Hausdorff case

Theorem 6.

Proof.

Putσ =τ|_H.

Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω. Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character. Theorem 3implies thatGr has aG_δ-diagonal of infinite rank. Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Theorem 6.

Proof.

Putσ =τ|_H. Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω.

Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character. Theorem 3implies thatGr has aG_δ-diagonal of infinite rank. Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Theorem 6.

Proof.

Putσ =τ|_H. Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω. Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character.

Theorem 3implies thatGr has aG_δ-diagonal of infinite rank. Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Theorem 6.

Proof.

Putσ =τ|_H. Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω. Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character. Theorem 3implies thatGr has aG_δ-diagonal of infinite rank.

Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Theorem 6.

Proof.

Putσ =τ|_H. Since(H, σ)has countableπ-character,(H, σr)too. By Lemma 5,πχ(H, τ_r|_H)≤ω. Since(H, τ_r|_H)is a dense subgroup of the regular spaceG_r, the regular paratopological groupG_r has countable π-character. Theorem 3implies thatGr has aG_δ-diagonal of infinite rank. Sinceτ_r ⊆τ, the space(G, τ)has aG_δ-diagonal of infinite rank.

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The Hausdorff case

Corollary and example

Corollary 7.

Every Hausdorff paratopological group of countableπ-character has a G_δ-diagonal of infinite rank.

If a Hausdorff topological groupGcontains a dense subgroup of countableπ-character, thenGis first countable.

The following example shows that, indeed,Theorem 6is more general thanCorollary 6.

Example 8.

There exists a Hausdorff paratopological group with uncountable π-character which contains a second countable dense subgroup.

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The Hausdorff case

Corollary and example

Corollary 7.

Example 8.

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The Hausdorff case

Corollary and example

Corollary 7.

Example 8.

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The Hausdorff case

Corollary and example

Corollary 7.

Example 8.

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The Hausdorff case

Corollary and example

Corollary 7.

Example 8.

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The Hausdorff case

Cardinal invariants of paratopological groups