Given that $X$ is a topological space with infinite cardinality and $X$ is homeomorphic to $X\times X$.  (A)   $X$ cannot be  connected.  (B)   $X$ cannot be  compact.  (C)   $X$ cannot be homeomorphic to a subspace of $\mathbb R$.  (D)   None of the above.    If I take $\mathbb Q$ with discrete topology then $\mathbb Q$  satisfies  the  given condition  while contradicting  option (C). Now  for  options  (A) and (B), I know if $X$ is connected (compact) then $X\times X$ will be connected (compact). But I cannot  think  of  any examples  of  connected  or  compact  spaces  that  are  homeomorphic  to  their  two-fold  product. Can anyone  provide with  some  examples?
Question is :  Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:   $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above.   I guess first two options are false.  We do have possibility that product of two connected spaces is connected.  So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.  We do have possibility that product of two compact spaces is compact.  So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.  I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.  So, only thing I have problem with is third option.  I could do nothing for that third option..  I would be thankful if some one can help me out to clear this.  Thank you :)
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RTL8101/2/6E PCI Express Fast/Gigabit Ethernet controller [10ec:8136] (rev 07)   output of lspci -knn | grep Net -A2  rahul@rahul-Inspiron-3542:~$ lspci -knn | grep Net -A2 06:00.0 Network controller [0280]: Qualcomm Atheros QCA9565 / AR9565 Wireless Network Adapter [168c:0036] (rev 01)     Subsystem: Dell QCA9565 / AR9565 Wireless Network Adapter [1028:020c]     Kernel driver in use: ath9k     Kernel modules: ath9k   output of sudo rfkill list  rahul@rahul-Inspiron-3542:~$ sudo rfkill list [sudo] password for rahul:  0: dell-rbtn: Wireless LAN     Soft blocked: yes     Hard blocked: yes 1: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no 2: hci0: Bluetooth     Soft blocked: yes     Hard blocked: no   output of sudo lshw -C network  rahul@rahul-Inspiron-3542:~$ sudo lshw -C network   *-network DISABLED                description: Wireless interface        product: QCA9565 / AR9565 Wireless Network Adapter        vendor: Qualcomm Atheros        physical id: 0        bus info: pci@0000:06:00.0        logical name: wlp6s0        version: 01        serial: 4c:bb:58:30:aa:07        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless        configuration: broadcast=yes driver=ath9k driverversion=4.17.11-041711-generic firmware=N/A latency=0 link=no multicast=yes wireless=IEEE 802.11        resources: irq:18 memory:f7900000-f797ffff memory:f7980000-f798ffff   *-network        description: Ethernet interface        product: RTL8101/2/6E PCI Express Fast/Gigabit Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:07:00.0        logical name: enp7s0        version: 07        serial: 74:e6:e2:14:2a:c8        size: 10Mbit/s        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=half firmware=rtl8106e-1_0.0.1 06/29/12 latency=0 link=no multicast=yes port=MII speed=10Mbit/s        resources: irq:19 ioport:e000(size=256) memory:f7800000-f7800fff memory:f2100000-f2103fff   *-network        description: Ethernet interface        physical id: 2        logical name: enp0s20u1        serial: 7a:14:5b:44:b4:96        capabilities: ethernet physical        configuration: broadcast=yes driver=rndis_host driverversion=22-Aug-2005 firmware=RNDIS device ip=192.168.42.109 link=yes multicast=yes   output of dmesg | grep ath9k  rahul@rahul-Inspiron-3542:~$ dmesg | grep ath9k [   23.712505] ath9k 0000:06:00.0 wlp6s0: renamed from wlan0   output of sudo iw dev  rahul@rahul-Inspiron-3542:~$ sudo iw dev [sudo] password for rahul:  phy#0     Interface wlp6s0         ifindex 3         wdev 0x1         addr 4c:bb:58:30:aa:07         type managed         txpower 0.00 dBm   output of dmesg   I already tried so many solutions but nothing works for me , I'm looking for answer from so many days. 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