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Interior of a Topological Space ... Singh, Proposition 1.3.2 (a) ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,882
Hobart, Tasmania
I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ...

I need help in order to formulate a rigorous proof of Proposition 1.3.2 (a) ... (using only the definition and results Singh has mentioned to date ... )

Proposition 1.3.2 reads as follows:


Singh - Propn 1.3.2 ... .png


Can some please help me to formulate a formal and rigorous proof of Proposition 1.3.2 (a) ... that is to formally and rigorously demonstrate that \(\displaystyle (A^{ \circ })^{ \circ } = A^{ \circ }\) ... using only Definition 1.3.1 ...


Help will be much appreciated ...

Peter
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,651
Leeds, UK
Can some please help me to formulate a formal and rigorous proof of Proposition 1.3.2 (a) ... that is to formally and rigorously demonstrate that \(\displaystyle (A^{ \circ })^{ \circ } = A^{ \circ }\) ... using only Definition 1.3.1 ...
This follows from the statement immediately before Proposition 1.3.2, where it says that $A$ is open if and only if $A = A^\circ$. Since $A^\circ$ is open, we can replace $A$ by $A^\circ$ in that statement, to conclude that $A^\circ = (A^\circ)^\circ$.