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... 3. Quasi semi-∗-normal spaces DEFINITION 3.1. A space (X, τ, I) is said to be quasi semi-∗normal if for any two disjoint π-closed sets A and B in (X, τ ), there exist disjoint semi-∗-open sets U, V such that A ⊆ U and B ⊆ V. DEFINITION 3.2. A space (X, τ, I) is said to be semi-∗-normal if for any tw ...
... 3. Quasi semi-∗-normal spaces DEFINITION 3.1. A space (X, τ, I) is said to be quasi semi-∗normal if for any two disjoint π-closed sets A and B in (X, τ ), there exist disjoint semi-∗-open sets U, V such that A ⊆ U and B ⊆ V. DEFINITION 3.2. A space (X, τ, I) is said to be semi-∗-normal if for any tw ...
LOCAL HOMEOMORPHISMS VIA ULTRAFILTER CONVERGENCE
... Proof. (i) ⇒ (ii) follows from Proposition 1. (ii) ⇒ (iii) is trivial, while to prove (iii) ⇒ (iv) we use Proposition 2 and the equality π1 · δf = 1X . From (iv) it follows that both f and δf are open maps, hence f is a local homeomorphism, by Proposition 1. Therefore the first four conditions are e ...
... Proof. (i) ⇒ (ii) follows from Proposition 1. (ii) ⇒ (iii) is trivial, while to prove (iii) ⇒ (iv) we use Proposition 2 and the equality π1 · δf = 1X . From (iv) it follows that both f and δf are open maps, hence f is a local homeomorphism, by Proposition 1. Therefore the first four conditions are e ...
mappings and decompositions of continuity on almost lindelöf spaces
... Moreover, it is clear that every strong θ-continuous mapping is θ-continuous and δcontinuous. It is clear also, that every δ-continuous mapping is almost δ-continuous but the converse is not true as Example 3.5 below shows. Contra-continuity implies subconracontinuity but the converse, in general, i ...
... Moreover, it is clear that every strong θ-continuous mapping is θ-continuous and δcontinuous. It is clear also, that every δ-continuous mapping is almost δ-continuous but the converse is not true as Example 3.5 below shows. Contra-continuity implies subconracontinuity but the converse, in general, i ...
Nonnormality of Cech-Stone remainders of topological groups
... α 7→ xα can be extended to a continuous map f : βω1 → E (here ω1 has the discrete topology), where E is the closure of F in βX. Claim 1. If q ∈ E \ X, then f −1 (q) is a single point which is contained in βω1 \ U (ω1 ). Indeed, let C be a closed neighborhood of q in βX that misses the compact set S0 ...
... α 7→ xα can be extended to a continuous map f : βω1 → E (here ω1 has the discrete topology), where E is the closure of F in βX. Claim 1. If q ∈ E \ X, then f −1 (q) is a single point which is contained in βω1 \ U (ω1 ). Indeed, let C be a closed neighborhood of q in βX that misses the compact set S0 ...
basic topology - PSU Math Home
... E XERCISE 1.1.3. On the integers Z, consider the profinite topology for which open sets are defined as unions (not necessarily finite) of arithmetic progressions (non-constant and infinite in both directions). Prove that this defines a topology which is neither discrete nor trivial. E XERCISE 1.1.4. ...
... E XERCISE 1.1.3. On the integers Z, consider the profinite topology for which open sets are defined as unions (not necessarily finite) of arithmetic progressions (non-constant and infinite in both directions). Prove that this defines a topology which is neither discrete nor trivial. E XERCISE 1.1.4. ...
