Topological Spaces, Filters and Nets Page: 4
i, 38 leavesView a full description of this thesis.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
of points interior to A is called the interior of A and is
denoted by
We note that the interior of a non-empty set can be
empty; this is the oase for a set reduced to a single point
when it is not open, for example in the real line.
Examples: In the plane consider the set Aw (z| ^ 2^
Now consider the monotonically decreasing sequence of closed
sets {B^ such that for each ot , is a closed disc with
center at the origin and each CZ A. This sequence of
closed sets will converge to the closed set which contains
only one point, namely the origin. The interior of a closed
set which contains only one point is empty.
For the discrete topology of the real line, each
single point is considered as an open set; therefore its
interior is a single point.
In the complex plane, consider the set A £z\[z\ < 1
and the point z « (Otl)j , A = J |z| £ 2] which does not
contain (0,l).
Definition 1.6 In a topological space E, a point x
is adherent to a set A if every neighborhood of x con-
tains at least one point of A. The set of points adherent
to A is called the adherence of A and is denoted by A,
(Note: In order that a set be closed, it is necessary
and sufficient that it be identical with its adherence.)
Proposition 1.2 If A is an open set in E, for every
subset B of E, A fl B CZ. Afl B.
Upcoming Pages
Here’s what’s next.
Search Inside
This thesis can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Thesis.
Cline, Jerry Edward. Topological Spaces, Filters and Nets, thesis, January 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130658/m1/7/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .