$$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

Analyzing Some Structural Properties of Topological B-Algebras

In this study, we investigate the topology on B-algebras: an algebraic system of propositional logic. We define here the notion of topological B-algebras (briefly, TB-algebras) and some properties are investigated. A characterization of TB-algebras based on neighborhoods is provided. We also provide a filterbase that generates a unique B-topology, making a TB-algebra in which the filterbase is a neighborhood base of the constant element, provided that the given B-algebra is commutative. Finally, we investigate subalgebras of TB-algebras and introduce the notion of quotient TB-algebras of the given B-algebra.

Klasifikasi Sinyal EEG Menggunakan Metode Fuzzy C-Means Clustering (FCM) Dan Adaptive Neighborhood Modified Backpropagation (ANMBP)

Instrumen EEG (electroencephalography) merupakan suatu instrumen yang digunakan sebagai perekam aktivitas otak dengan memperlihatkan gelombang otak. Prinsip kerja EEG adalah dengan mendeteksi perubahan muatan secara tiba-tiba dari sel neuron yang ditandai dengan adanya interictal spike-and-wave pada hasil EEG (electroencephalogram). Terdapat suatu data set sinyal EEG, direkam pada sukarelawan normal dan epilepsi. Pada penelitian ini dengan menggunakan data tersebut akan dilakukan suatu sistem klasifikasi sinyal EEG dengan berdasar pada kondisi normal dan epilepsi. Klasifikasi sinyal EEG menggunakan Metode Adaptive Neighborhood Base Modified Backpropagation (ANMBP). Hasil ekstraksi fitur dari sinyal EEG dengan menggunakan metode Fuzzy C-Means (FCM) Clustering, dimana proses awalnya melalui dekomposisi wavelet menggunakan Discrete Wavelet Transform (DWT) dengan level 2 didapatkan 3 koefisien wavelet kemudian pada masing masing koefisien tersebut di clustering menggunakan FCM dengan 2 cluster sehingga menghasilkan 6 fitur yang akan menjadi vektor fitur. Dari vektor fitur tersebut digunakan sebagai inputan untuk dilakukan proses klasifikasi dengan menggunakan metode ANMBP. Hasil sistem sementara didapatkan recognition rate sebesar 74.37%.

Free paratopological groups

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces $X$ for which the group FP(X) has the inductive limit property.

Neighborhood base at the identity of free paratopological groups

In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.

On the separately open topology

We consider the relationship between separately continuous functions and separately open sets, and we study the properties of the separately open topology on R2 and on Q2. We show that R2 with this topology (denoted R ⊗R) is completely and strongly Hausdorff and that Q⊗Q is normal but not a p-space. In addition, we show that each point of Q ⊗Q has an uncountable neighborhood base.

ON H-Sets and Open Filter Adherences(1)

The relationship between H-sets and open filter adhérences is considered. The open filter adhérences of an H-closed space are shown to be H-sets; and, a necessary and sufficient condition is given for an H-set S, of a Hausdorff space X, to be an open filter adherence. A necessary condition is determined for the existence of a minimal adherent set which contains S; and, in the case that X is H-closed, sufficient conditions are determined. As a related result, an H-closed space X is shown to be seminormal if every H-set of X possesses a neighborhood base consisting of regular open sets.

Absolute C-Embedding of Spaces with Countable Character and Pseudocharacter Conditions

Absolute C-embeddings have been studied extensively by C. E. Aull. We will use his notation P = C[Q] to mean that a space satisfying property Q is C-embedded in every space having property Q that it is embedded in if (and only if) it has property P. The first result of this type is due to Hewitt [5] where he proves that if Q is “Tychonoff” then P is almost compactness. Aull [2] proves that if Q is “T4 and countable pseudocharacter” or “T4 and first countable” then P is “countably compact”. In this paper we show that P is almost compactness if Q is “Tychonoff” and any of countable pseudocharacter, perfect, or first countability. Unfortunately for the last case we require the assumption that . Finally we show that P is countable compactness if Q is Tychonoff and “closed sets have a countable neighborhood base”. In each of the above results C-embedding may be replaced by C*-embeddings and the results hold if restricted to closed embeddings.