Exactly nn-resolvable Topological Expansions

For $\kappa$ a cardinal, a space X=(X,\sT) is $\kappa$-{\it resolvable} if $X$ admits $\kappa$-many pairwise disjoint \sT-dense subsets; (X,\sT) is {\it exactly} $\kappa$-{\it resolvable} if it is $\kappa$-resolvable but not $\kappa^+$-resolvable. The present paper complements and supplements the authors' earlier work, which showed for suitably restricted spaces (X,\sT) and cardinals $\kappa\geq\lambda\geq\omega$ that (X,\sT), if $\kappa$-resolvable, admits an expansion \sU\supseteq\sT, with (X,\sU) Tychonoff if (X,\sT) is Tychonoff, such that (X,\sU) is $\mu$-resolvable for all $\mu<\lambda$ but is not $\lambda$-resolvable (cf. Theorem~3.3 of \cite{comfhu10}). Here the "finite case" is addressed. The authors show in ZFC for $1<n<\omega$: (a) every $n$-resolvable space (X,\sT) admits an exactly $n$-resolvable expansion \sU\supseteq\sT; (b) in some cases, even with (X,\sT) Tychonoff, no choice of \sU is available such that (X,\sU) is quasi-regular; (c) if $n$-resolvable, (X,\sT) admits an exactly $n$-resolvable quasi-regular expansion \sU if and only if either (X,\sT) is itself exactly $n$-resolvable and quasi-regular or (X,\sT) has a subspace which is either $n$-resolvable and nowhere dense or is $(2n)$-resolvable. In particular, every $\omega$-resolvable quasi-regular space admits an exactly $n$-resolvable quasi-regular expansion. Further, for many familiar topological properties \PP, one may choose \sU so that (X,\sU)\in\PP if (X,\sT)\in\PP

Tychonoff Expansions with Prescribed Resolvability Properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and Garc\'ia-Ferreira (2001): (1) Is every $\omega$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably restricted Tychonoff topological space (X,\sT) admits a larger Tychonoff topology (that is, an "expansion") witnessing such failure. Specifically the authors show in ZFC that if (X,\sT) is a maximally resolvable Tychonoff space with S(X,\sT)\leq\Delta(X,\sT)=\kappa, then (X,\sT) has Tychonoff expansions \sU=\sU_i ($1\leq i\leq5$), with \Delta(X,\sU_i)=\Delta(X,\sT) and S(X,\sU_i)\leq\Delta(X,\sU_i), such that (X,\sU_i) is: ($i=1$) $\omega$-resolvable but not maximally resolvable; ($i=2$) [if $\kappa'$ is regular, with S(X,\sT)\leq\kappa'\leq\kappa] $\tau$-resolvable for all $\tau<\kappa'$, but not $\kappa'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.Comment: 25 pages, 0 figure

Topological partition relations to the form omega^*-> (Y)^1_2

Theorem: The topological partition relation omega^{*}-> (Y)^{1}_{2} (a) fails for every space Y with |Y| >= 2^c ; (b) holds for Y discrete if and only if |Y| <= c; (c) holds for certain non-discrete P-spaces Y ; (d) fails for Y= omega cup {p} with p in omega^{*} ; (e) fails for Y infinite and countably compact

Evaluation of the Status of Guidance Services in Secondary Schools in Rivers State, Nigeria

The study investigated the status of guidance services in some Secondary schools in Rivers State. Out of the two hundred and fifty (250) public schools in Port Harcourt, only twenty (20) have at least a practicing counsellor. Fourteen schools with practicing counsellors were sampled in this study. The two Federal Government and University Demonstration Schools in Port Harcourt were also included in the study. Data was collected through the administration of a questionnaire on guidance services. Data analysis revealed Guidance and Counselling services in Rivers State are faced with many obstacles which jeopardize effective delivery of the services, thereby resulting in educational wastage instead of the expected quality outcome. Funds for provision of basic guidance services are lacking. Most of the vital counselling services needed for quality education are the least regularly performed in schools e.g. use of psychological tests and keeping of cumulative record folders. Only 18% of the schools sampled indicated that the objectives of the National Policy on education with respect to guidance and counselling services have been realized to some extent. If the stated objectives of Education in the National Policy on Education are to be achieved, the current status of guidance services needs to be reviewed and positive steps taken towards its improvement. Keywords: Evaluation, status, guidance services

Counselling Blind-Impaired Adolescents in Their Self-Perception of Social Relationship

A social activity scale was administered to 220 mainstreamed blind-impaired adolescents. In general, students reported participating in school activities more frequently with blind-impaired than with blind peers, but this was qualified by the extent that the students were mainstreamed. Ratings of participation with blind-impaired peers, decreased for the students who were mainstreamed for more classes, Students indicated that they were more emotionally secure with blind-impaired peers, and there was no increase in emotional security with blind peers with more mainstreaming. Responses of students who were mainstreamed for more classes suggested that they realized they had less interaction with blind-impaired peers even though this was the group with whom they were most comfortable. Although these students were surrounded with blind peers, this contact did not appear to promote identification and relational bonds with them. When there is opportunity for participation, the quality of the relationships is not necessarily positive. Keywords: Counselling, Self perception, Social Relationship

Counseling Students on Choice of Careers in Professional Areas in Rivers State University of Science and Technology, Port Harcourt: Implication for Counselling

The study investigated final year students of Faculty of Technical and Science Education and third year students of Faculty of Sciences choice of career in 2011/2012 academic session in Rivers State University of Science and Technology, Port Harcourt. The instrument used for data collection was a self designed questionnaire administered on 260 student, 130 subjects from the Faculty of Technical and Science Education and 130 subjects from the Faculty of Sciences. Rivers State University of Science and Technology, Port Harcourt. Survey design was utilized, t-test statistics was used to test the three null hypothesis at 0 05 level of significance. Results showed that hypotheses one and three were not significant at 0.05 level of significance, hence they were accepted. Hypothesis two was significant at 005 level of significance, it was therefore rejected. The implications of the findings were also discussed. Keywords: Counselling, career choice, Professional areas