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Closed Subsets in a Toplogical space ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
I am reading Sasho Kalajdzievski's book: "An Illustrated Introduction to Topology and Homotopy" and am currently focused on Chapter 3: Topological Spaces: Definitions and Examples ... ...

I need some help in order to fully understand Kalajdzievski's definition of a closed set in a topological space ...


The relevant text reads as follows:


K - Defn of a Closed Subset of a Toplogical Space ... .png


As I understand it many closed subsets of the underlying set \(\displaystyle X\) of a topological space \(\displaystyle (X, \tau)\) do not belong to the topological space because they are not open ... i.e. they are not clopen sets ...

Is my interpretation of the above situation correct ... ... ?


Help will be appreciated ...

Peter

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It may help readers of the above post to have available Kalajdzievski's definition of a topological space ... so I am providing the same ... as follows:


K - Defn of a Topological Space ... .png
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
As I understand it many closed subsets of the underlying set [FONT=MathJax_Math]XX[/FONT] of a topological space [FONT=MathJax_Main]([FONT=MathJax_Math]X[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]τ[/FONT][FONT=MathJax_Main])[/FONT](X,τ)[/FONT] do not belong to the topological space because they are not open ... i.e. they are not clopen sets …


I would say "they do not belong to the topology" rather than that "they do not belong to the topological space". The "topology" is the collection of open subsets of X while the "topological space" is X together with a given topology.

But yes, the closed sets do not necessarily belong to the topology- closed sets are not necessarily open sets. In fact, in most examples of topologies, such as the "usual metric topology" on the real numbers, closed sets are sharply distinguished from open sets.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
I would say "they do not belong to the topology" rather than that "they do not belong to the topological space". The "topology" is the collection of open subsets of X while the "topological space" is X together with a given topology.

But yes, the closed sets do not necessarily belong to the topology- closed sets are not necessarily open sets. In fact, in most examples of topologies, such as the "usual metric topology" on the real numbers, closed sets are sharply distinguished from open sets.
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Thanks HallsofIvy ... appreciate your help ...

Peter