#### Date of Submission

2-28-1972

#### Date of Award

2-28-1973

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Quantitative Economics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Mitra, Ashok (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

Given a topological property Ï€ and a pon-void set X, let Ï€(X) denote the set of topologies on X with property Ï€. 7 (X) is obviously partially ordered by inclusion, A topologi- cal space (X,T) is minimal if T is a minimal element in T (X). (X, T) is said to be maximal Ï€ if T1 1s a maximalelement in Ï€(X), The study of maximal and minimal topological space is, as a matter of fact, a study of maximal Ï€ and minimal Ï€ spaces. Topological spaces closely related to minimal T sp aces are T-closed and KatetoV T spaces, (X,T) 1s T-closed if T has property T and X is a closed subspace of every T sp ace in which it can be embedded, A Ï€ space (X, T) is KatÅ¡tov I provided T is stronger than some minimal T topology on X,The study of maximal topolo gies owes its origin to the remarkable fact that a compact Hausdo rff space is maximal co mp ac t. This was first observed by R, Vaidyanathaswamy in his book [Valpublished ir the year 1947. The first substantial wo rk in maximal topologies appeared in 1963 when N. Smythe and C. A, Wilkins characterised maximal compact spaces [SW]. In this brilliant p aper they also produced an example of a maximal compact space which is not Hausdorff. It was followed up by J.P.Thomas in 1968 when he studied maximal conneefed spaces [Thl. In Chapter I of this thesis, as promised by the title of the chapter, we proceed to investigate maximal r spaces for some further topological properties 1ike H-closed, connected, lightly compact, pseudo oomp ac t, LindelÃ¶f and countably compact. paucity of published literature on maximal topologies is largely due to the simple reason that for ost topological properties the maximal spaces turn out to be discrete. The impetus to study ninimal topological space came from the well-known topological result, first observed by A, S. Parkhomenko [Pa] in 1939, that a compact Hausdo rff space is minimal Hausdo rff. Earlier in 1924 P.3. Alexandroff and P. Urysohn [AU] investigated H-closed' spaces and proved that a regular space is H-closed if and only if it is compact. In considering the I-elosedness of a Hausdo rff space they were guided by the important observation that if a compact Hausdorff space X is embedded in a Hausdo rff space Y, the inage of X is always closed in Y, Parkhomenko found out the relationship between minimal Hausdorff and H-elosed spaces by denonstrating that a minimal Hausdorff space is always H-closed. It was left to Katetov (K] to obtain the characteri- sations of ninimal Hausdo rff spaces, In 1941 E, Cartan [Bol) could obtain characterisations of bo th minimal Hausdo rff and -elosed spaces in terms of filters, He is the first to produce a non-ompact minimal Hausdo rff space, Since then sinimal w and r-clesed spaces have been studied fora wide spectrum of topological properties 1ncluding various separation properties. In chapter II of this thesis we carry out further studies in minimal Ï€ and -closed spaces for Ï€ = realcompact, first countable realcompact, locally H-closed, E1 space, P-space, Hausdorff P-space, analytic and borelian (Relevant definitions are supplied later in this thesis).

#### Control Number

ISILib-TH10

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Raha, Asit Baran Dr., "Maximal and Minimal Topologies." (1973). *Doctoral Theses*. 326.

https://digitalcommons.isical.ac.in/doctoral-theses/326

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843414