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

#### Peter

##### Well-known member
MHB Site Helper
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 ... )

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
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 $A^\circ = (A^\circ)^\circ$.