# which of the following is $T_1$ space?

From Topology without tears

A Topological space $$(X,\tau)$$ is said to be $$T_1$$ space if every singleton set is closed in $$(X,\tau)$$. Show that precisely two of the following 9 topological spaces are $$T_1$$.

1.a discrete space

(I believe discrete is $$T_1$$ because every subset in discrete space is open and closed both.So singleton are also closed.)

2.An indiscrete space with atleast two points

(Since its topology contains only $$X$$ and $$\phi$$ I believe singleton will not be neither closed nor open.)

3.An infinite set with cofinite topology

(I have a example $$\tau$$ consisting of $$\mathbb{N}$$,$$\phi$$ and every set $$\{n,n+1,...\}$$ for any positive integer.Here $$\mathbb{N}$$ is infinite set and this is cofinite topology since every closed set will be finite here.But here singletons need not be closed.)

4.$$X=\{a,b,c,d,e,f\}$$ and $$\tau=\{ X,\phi,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$$

(Here I can see every singleton are not closed.)

5.$$\tau_1$$ consist of $$\mathbb{R},\phi$$, and every interval $$(-n,n)$$ for any $$n$$ positive integer.

(for $$\{n\}$$ to be closed its complement $$(-\infty,-n] \cup [n,\infty)$$ must belong to $$\tau_1$$.Which is not the case.So $$\tau_1$$ is not $$T_1$$)

1. $$\tau_2$$ consist of $$\mathbb{R},\phi$$, and every interval $$[-n,n]$$ for any $$n$$ positive integer.

(Similarly I can show $$\tau_2$$ not $$T_1$$)

7.$$\tau_3$$ consist of $$\mathbb{R},\phi$$, and every interval $$[n,\infty)$$ for any $$n$$ positive integer.

(Similarly $$\tau_3$$ is also not $$T_1$$.

8.$$\tau_1$$ consists of $$\mathbb{N}$$,$$\phi$$ and every set $$\{1,2,...,n\},$$for $$n$$ any positive integer.

(Here also all singleton will not be closed.because its complement need not be in $$\tau_1$$)

9.$$\tau_2$$ consists of $$\mathbb{N}$$,$$\phi$$ and every set $$\{n,n+1,...\},$$for $$n$$ any positive integer.

(Here also all singleton will not be closed.because its complement need not be in $$\tau_1$$)

So,I got only one $$T_1$$ space.where I am wrong?

• Minor error: In $(5)$ the complement of $\{n\}$ is $(-\infty,n)\cup(n,\infty)$. Jan 11, 2021 at 14:27
$$3$$ is also a $$T_1$$ space. The complement of singleton $$\{a\}$$ is $$X-\{a\}$$ which is open since its complement $$\{a\}$$ is finite (a non-empty set is open in $$\color{red}{\text{co}}$$finite topology iff its complement is finite).
The example you have taken is not the cofinite topology on $$\Bbb N$$. It is a subset (strictly coarser) of the cofinite topology. Note that $$\Bbb N-\{2\}\in$$ the cofinite topology on $$\Bbb N$$ but not your example topology.