- #1

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- Homework Statement
- Standard topology is coarser than lower limit topology?

- Relevant Equations
- T={u subset R: for all x in u exists d>0 s.t. (x-d,x+d) subset u}

Hello everyone,

Our topology professor have introduced the standard topology of ##\mathbb{R}## as:

$$\tau=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left(x-\delta,x+\delta\right)\subset u\right\},$$

and the lower limit topology as:

$$\tau_{1}=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left[x,x+\delta\right)\subset u\right\}.$$

He asked for the relation between the two topologies. It is easy to show that ##\tau_{1}## is coarser than ##\tau## according to this definition:

$$\left(x-\delta,x+\delta\right)\subset \left[x,x+\delta\right)\subset v,$$

for ##v## in ##\tau_{1}##. But that is not true in all topology references that I read. According to their definition (the collection of all open intervals in the real line forms a standard topology) the standard topology is coarser than lower limit topology.

Will appreciate any help, thanks.

Our topology professor have introduced the standard topology of ##\mathbb{R}## as:

$$\tau=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left(x-\delta,x+\delta\right)\subset u\right\},$$

and the lower limit topology as:

$$\tau_{1}=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left[x,x+\delta\right)\subset u\right\}.$$

He asked for the relation between the two topologies. It is easy to show that ##\tau_{1}## is coarser than ##\tau## according to this definition:

$$\left(x-\delta,x+\delta\right)\subset \left[x,x+\delta\right)\subset v,$$

for ##v## in ##\tau_{1}##. But that is not true in all topology references that I read. According to their definition (the collection of all open intervals in the real line forms a standard topology) the standard topology is coarser than lower limit topology.

Will appreciate any help, thanks.