Caryn Navy | |
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Born | |
Nationality | American |
Spouse | David Holladay |
Academic background | |
Education | Massachusetts Institute of Technology |
Alma mater | University of Wisconsin–Madison |
Thesis | Nonparacompactness in Para-Lindelöf Spaces (1981) |
Doctoral advisor | Mary Ellen Rudin |
Academic work | |
Discipline | Mathematics Computer science |
Sub-discipline | Set-theoretic topology Braille technology |
Caryn Linda Navy (born July 5,1953) is an American mathematician and computer scientist. Blind since childhood,she is chiefly known for her work in set-theoretic topology and Braille technology.
Navy was born in Brooklyn,New York in 1953. Born premature,she was diagnosed as totally blind from retinopathy of prematurity. Her family soon discovered that she could actually see from the corner of one eye,but at age 10 she lost all sight due to retinal detachment. [1]
The next year,in sixth grade,Navy began learning to read and write Braille at school. She also learned the Nemeth Braille system for writing mathematics, [2] which became her favorite subject. She enjoyed team math competitions,and at age 14 independently rediscovered Euclid's formula for even perfect numbers. [1] She also learned Hebrew Braille in preparation for her bat mitzvah service. [3] At age 16 Navy was hired for her first job,as a Dictaphone typist in New York City. She took a class to learn to travel the New York City Subway. [1]
Navy attended the Massachusetts Institute of Technology 1971–1975,majoring in mathematics. The only textbook she had in Braille was her calculus book. All her other books were obtained as audiobooks from Recording for the Blind. At MIT,her undergraduate advisor James Munkres introduced her to the subject of topology. Upon her graduation with a bachelor's degree in mathematics in 1975, [2] she received the AMITA Senior Academic Award from the Association of MIT Alumnae. [4] Early in her undergraduate career,Navy met David Holladay,an electrical engineering student. He looked up enough Braille to write her a note after their first meeting. They were married after graduation. [1]
Navy attended graduate school at the University of Wisconsin–Madison,majoring in mathematics,with a minor in computer science. During her graduate education,she used an Optacon device to read textbooks that were not available in Braille or as audiobooks. [1] She received her M.A. in 1977, [2] and her Ph.D. in 1981 under the supervision of topologist Mary Ellen Rudin. [5]
Navy's doctoral thesis,"Nonparacompactness in Para-Lindelöf Spaces",was important in the development of metrizability theory. The paper examines the properties of para-Lindelöf topological spaces,which are a generalization of both Lindelöf spaces and paracompact spaces. In a para-Lindelöf space,every open cover has a locally countable open refinement,that is,one such that each point of the space has a neighborhood that intersects only countably many elements of the refinement. The spaces constructed by Navy are counterexamples to the conjecture that all para-Lindelöf spaces are paracompact. Some of her spaces are even normal Moore spaces under suitable set-theoretic assumptions. Since every metrizable space is paracompact,these are counterexamples to the normal Moore space conjecture.
Stephen Watson called Navy's construction "a rather general one that permitted quite a lot of latitude" and said,"No other way of getting para-Lindelöf is known. I don't think another way of getting para-Lindelöf is even possible—Navy's method looks quite canonical to me." [6] In 1983,William Fleissner modified one of her spaces to be a normal Moore space under the assumption of a particular covering property. Fleissner's examples finally resolved the normal Moore space conjecture by showing that it requires large cardinal axioms. [6] [7]
After graduate school,Navy took a position as a visiting assistant professor in the mathematics department of Bucknell University in 1981. [2] While they were living in Lewisburg,Pennsylvania,Holladay started a software company called Raised Dot Computing,focused on computer Braille technology. He was inspired by Navy's need for Braille translation to help with her math teaching. [8] She was an important consultant for the company,helping with software ideas and testing products in her university work. In December 1981,Raised Dot Computing released its first major product,BRAILLE-EDIT,a word processor and two-way Braille translator program for the Apple II. [9] Navy left the university in 1984 to work for the company, [2] and the couple and the company moved back to Madison,Wisconsin in July 1984. [9]
Navy applied her computer skills to improving Raised Dot Computing's assistive software,producing enhanced versions of BRAILLE-EDIT and utility programs to handle textbook-format Braille and other special formats. In 1985 they added a line of MS-DOS software,culminating in the company's most successful product,MegaDots. Released in August 1992 and maintained until 2016,MegaDots provided Braille translation and word processing for the PC. Raised Dot Computing was transferred in September 1998 to a Wisconsin nonprofit organization called Braille Planet, [9] which was then bought out in August 1999 by Duxbury Systems. [10] Navy and Holladay live in Westford,Massachusetts, [8] and still work at Duxbury. [11]
Navy is a member of the Braille Authority of North America,where she serves as a consultant to the Nemeth Code Technical Committee. [12]
In topology and related areas of mathematics,a metrizable space is a topological space that is homeomorphic to a metric space. That is,a topological space is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology,the focus here is on general topology. The following definitions are also fundamental to algebraic topology,differential topology and geometric topology.
In topology and related branches of mathematics,a normal space is a topological space X that satisfies Axiom T4:every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces,or T5 spaces,and perfectly normal Hausdorff spaces,or T6 spaces.
In mathematics,a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné(1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal,and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
In mathematics,the lower limit topology or right half-open interval topology is a topology defined on ,the set of real numbers;it is different from the standard topology on and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [a,b),where a and b are real numbers.
In topology,the long line is a topological space somewhat similar to the real line,but in a certain way "longer". It behaves locally just like the real line,but has different large-scale properties. Therefore,it serves as an important counterexample in topology. Intuitively,the usual real-number line consists of a countable number of line segments laid end-to-end,whereas the long line is constructed from an uncountable number of such segments.
In topology,a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete,anti-discrete,concrete or codiscrete. Intuitively,this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.
In topology,the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line,which is the real line under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.
In topology,a second-countable space,also called a completely separable space,is a topological space whose topology has a countable base. More explicitly,a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms,the property of being second-countable restricts the number of open sets that a space can have.
In topology and related areas of mathematics,a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively,a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is,a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally,a topological property is a property of the space that can be expressed using open sets.
In topology,a branch of mathematics,a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces,with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold,obtained by simply "forgetting" the added structure. However,not every topological manifold can be endowed with a particular additional structure. For example,the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
Mary Ellen Rudin was an American mathematician known for her work in set-theoretic topology. In 2013,Elsevier established the Mary Ellen Rudin Young Researcher Award,which is awarded annually to a young researcher,mainly in fields adjacent to general topology.
In mathematics,in the field of topology,a topological space is said to be pseudonormal if given two disjoint closed sets in it,one of which is countable,there are disjoint open sets containing them. Note the following:
In mathematics,more specifically point-set topology,a Moore space is a developable regular Hausdorff space. That is,a topological space X is a Moore space if the following conditions hold:
In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
In mathematics,a topological space is called collectionwise normal if for every discrete family Fi (i ∈I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈I),such that Fi ⊆Ui. Here a family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from . An equivalent definition of collectionwise normal demands that the above Ui (i ∈I) themselves form a discrete family,which is stronger than pairwise disjoint.
The Morita conjectures in general topology are certain problems about normal spaces,now solved in the affirmative. The conjectures,formulated by Kiiti Morita in 1976,asked
In mathematics,particularly topology,a Gδ space is a topological space in which closed sets are in a way ‘separated’from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces,and satisfy the strongest of separation axioms.
David Holladay was an American computer programmer who worked on early Braille translator word processing software allowing blind Apple Computer users to enter,edit,and translate text.